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ALGEBRA; 


ADAPTED   rO  THE 


COURSE  OF  INSTRUCTION  USUALLY  PURSUED 


IN  THE 


€o\Up$  miA  ^ailf  mwis 


OF  THE 


UNITED     STATES. 


BY 


P.   A.   TOWNE, 

// 

FORMERLY   GENERAL    PRINCIPAL   OP   THE   BARTON    ACADEMY,    MOBILE,    ALA. 
PROFESSOR   OF    MATHEMATICS,    CLINTON    LIBERAL    INSTITUTE,    N.    Y. 


LOUISVILLE,  KY.: 
JOHN    P.   MORTON    &    CO. 


\53 


^^5 


cc«:       cc      c*c 


%lf  y  >  ^  r^^^^a:-^'^^ 


Entered,  according  to  Act  of  Congress,  in  the  year  18C5,  by 

JOHN  P.  MOllTON  &  CO. 

in  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  District  of  Kentucky. 


ZLECTROTTPED  BY  L.  J0IIX30N   &   CO. 
PHIIAm'T.pni*. 


PREFACE. 


It  would  be  easy  to  state,  by  way  of  preface,  the  pre- 
cise reasons  which  have  led  the  author  to  add  this 
Algebra  to  the  numberless  treatises  on  the  same  subject 
already  in  existence.  If,  however,  the  reader  will  con- 
sent to  devote  a  few  leisure  moments  to  an  examination 
of  the  following  points,  the  writer  flatters  himself  with 
the  belief  that  these  reasons  v\-ill  appear  much  more 
forcibly  than  if  stated  in  the  language  of  an  argument 

Attention  is  invited  to  the  accuracy  of  the  Definitions; 
to  the  brevity  and  clearness  of  the  demonstrations;  the 
explanation  of  i^osiiive  and  negative  quantities;  the  subject 
oi  factoring ;  the  appropriateness  and  careful  gradation  of 
equations  and  other  problems ;  the  manner  in  which  the 
transition  from  the  reduction  of  equations  to  the  solution 
of  problems  is  effacted ;  the  perpetual  recurrence  of  the 
mind  of  the  pupil,  as  he  advances,  to  first  principles, — as, 
for  instance,  compare  the  subjects  of  Greatest  Common 
Divisor,  Least  Common  Multiple,  Eeduction  of  Fractions 
and  Equations  having  equal  roots;  to  the  circumstance 
that  many  of  the  problems  have  been  doubled  by  placing 
figures  in  parentheses  corresponding  to  each  other;  the 
constant    requirement    of    reducing    generalizations    to 

3 

O O  «^  v>-  "IS:  '^ 


4  PREFACE. 

numerical  problems  previously  solved ;  the  treatment  of 
Quadratic  Equations,  particularly  those  involving  two 
unknown  quantities;  the  subject  of  Logarithms;  and, 
finally,  to  the  practical  manner  in  which  the  Higher 
Equations  are  treated. 

Many  other  points  might  be  mentioned ;  but,  in  pass- 
ing over  the  above,  the  reader  will  not  fail  to  discover 
them. 

In  the  preparation  of  the  work,  the  author  has  been 
occupied  some  ten  or  twelve  years;  and  he  now  feels  safe 
in  pledging  himself  that  no  material  change  will  be  made 
in  any  of  its  discussions  during,  at  least,  the  same  length 
of  time.  Every  part  of  it  has  been  repeatedly  tested  in 
the  class-room. 


CONTENTS. 


CHAPTER  I. 

PAGE 

Definitions  and  Exercises 9 

Symbols  of  Quantity 9 

of  Operation 10 

of  Exponents 10 

of  Coefficients 11 

of  Relation 12 

Examples 14 

Notation 17 

CHAPTER  11. 

Addition 18 

Subtraction 24 

Multiplication 32 

Division 42 

CHAPTER  III. 

Factoring 51 

Greatest  Common  Divisor 55 

Least  Common  Multiple 58 

General  Review 50 

CHAPTER  IV. 

Fractions 61 

Review  of  Fractions 75 

1*  5 


6  CONTENTS. 

CHAPTER  V. 

PAGE 

Equations  of  the  First  Degree 76 

Literal  Equations 86 

Problems 88 

Equations  of  the  First  Degree  involving  Two  Unknown  Quantities 98 

Elimination C3 

by  Addition  or  Subtraction 00 

by  Substitution 104 

by  Comparison 108 

Three  Equations  involving  Three  Unknown  Quantities Ill 

Four  or  more  Equations  involving  a  like  number  of  Unknown  Quan- 
tities    112 

Symmetrical  Equations 113 

Problems  involving  Two  or  more  Unknown  Quantities i llo 

Literal  Equations 121 

Generalizations 122 

Negative  Ptesults •••  125 

Interpretation  of  the  Symbols    -'  ^'  -' 129 

Indeterminate  Analysis 131 


CHAPTER  YI. 

Involutions 133 

Logarithms 1^2 

Multiplication  by  Logarithms 152 

Division  by  Logarithms 153 

Arithmetical  Complement 154 

Involution  by  Logarithms 155 

Extraction  of  Roots  by  Logarithms 156 

Evolution  and  Treatment  of  Radicals 159 

Imaginary  Quantities 1"2 

CHAPTER  VII. 

Equations  of  the  Second  Degree 1"6 

Problems  producing  Incomplete  Equations  of  the  Second  Degree 178 

Complete  Equations  of  the  Second  Degree ISO 


CONTENTS.  7 

PAGE 

Trinomial  Equutious 187 

Literal  Equations , 189 

Problems  involving  Equations  of  the  Second  Degree 194 

Equations  with  Two  Unknown  Quantities 197 

Review 202 

Homogeneous  Equations 20'J 

Equations  containing  Three  Unknown  Quantities 211 

Problems  involving  Two  Unknown  Quantities 212 

General  Properties  of  Equations  of  the  Second  Degree 218 

Application  of  these  Properties 210 

Ratio  and  Proportion 222 

Problems 227 

Arithmetical  Progression 229 

Geometrical  Progression 234 

Problems 237 

Indeterminate  Coefficients 239 

Binomial  Theorem 213 

The  Table  of  Logarithms 244 

Practical  Applications 240 

CHAPTER  VIII. 

Equations  of  the  Third  Degree 251 

Law  of  Derived  Polynomials 205 

Properties  of  Derived  Polynomials 205 

Equal  Roots 200 

General  Solution  of  the  Equation  of  the  Third  Degree 208 

Numerical  Solution  of  Cubic  Equations 200 

Higher  Equations 270 

Questions  for  Examination 279 


'>.-Ai.Ji''0^'^iUj/;i 


AL  GEB  E  A. 


CHAPTEE    I. 

DEFINITIONS   AND    EXERCISES. 

1.  Algebra  investigates  the  relations  of  quantities  by  Symbols. 

SYMBOLS    OF   QUANTITY. 

2.  The  symbols  of  quantity  in  Algebra  are  the  letters  of  the 
alphabet. 

3.  The  first  letters  of  the  alphabet,  viz.,  a,  6,  c,  .  .  . ,  usually 
represent  quantities  whose  numerical  values  are  Icnoivn, 

4.  The  last  letters,  viz.,  x,  y,  z^  represent  quantities  whose 
values  are  unknown,  —  /.  e.,  unknown  before  the  operations  in 
which  they  are  involved  are  performed ;  after  these  operations 
unhnown  quantities  become  known, 

5.  An  algebraic  quantity  is  properly,  then,  a  quantity  repre- 
sented by  a  letter  or  letters. 

6.  An  arithmetical  quantity  is  one  represented  by  a  figure  or 
figures. 

•Y.  Algebraic  quantities  are  therefore  called  literal  quantities,  to 
distinguish  them  from  numerical  quantities.  Both  kinds  of  quan- 
tities are  used  in  Alojebra. 


1 0  I)  ]•:  F I :;  i  i  i  c  :.*  .s    and    e  x  e  k  c  i  s  e  s  . 

SYMBOLS    OF    OPERATION. 

8.  The  sign  +)  plus,  indicates  that  the  quantity  before  whicli 
it  is  placed  is  to  be  taken  additively.  Thus,  a  -\-h,  a  plus  h, 
denotes  that  the  quantity  h  is  to  be  added  to  the  quantity  a. 

9.  The  sign  — ,  minus,  indicates  that  the  quantity  before 
whicli  it  is  placed  is  to  be  taken  suhtractively.  Thus,  a  —  h, 
a  minus  h,  denotes  that  the  quantity  h  is  to  be  subtracted  from 
the  quantity  a. 

When  no  sign  is  written,  +  is  understood.  Thus,  a  is  the 
same  as  +  a. 

10.  There  are  three  ways  in  which  to  indicate  Multiplication 
in  Algebra,  viz.,  o:Xy,  x.y,  and  xy,  all  of  which  indicate  that 
X  is  to  be  multiplied  by  y, 

11.  There  are  also  three  ways  in  which  to  indicate  Division 
in  Algebra,  viz.,  x~ry,  ~,  and  x\l,  all  of  which  signify  that  x 
is  to  be  divided  by  y. 

12.  The   sign    (      1,   parenthesis,   or  ,    vinculum,   whicli 

may  also  be  drawn  perpendicularly,  is  used  to  connect  several 
algebraic  expressions,  and  denotes  that  they  are  to  be  treated 
as  a  single  expression.     Thus, 

-{.  h  is  the  same  as  a  -\-h  -{-  c.x,  or  (a -\- h -{- c')x. 

+  c 
all  of  which  signify  that  the  sum  of  a,  h,  and  c  is  to  be  multi- 
plied by  X.     Again,  (4  +  5)  x  6  is  the  same  as  54,  but  4  -f  5  x  0 
is  the  same  as  34. 

13.  Of  Exponents. — In  the  expressions  a.  h  c;  x  y  z;  m  n; 
etc.,  each  of  the  letters  composing  the  expression  is  c.rod  a 
literal  factor.  If  a  letter  is  to  occr.r  ;i«  a  factor  sevci;;l  times, 
instead  of  writing  a  a  ;  x  x  x;  y  y  y  y;  etc.,  a  figure   is  placed 


DEFINITIONS     AND      EXERCISES.  11 


at  the  right  hand  of  the  letter  and  a  little  above;  thus,  a^;  x^;  y^\ 
etc.,  signify  that  ic,  y,  and  ;;  have  been  taken  as  factors,  twice, 
three  times,  four  times,  etc.  This  figure  is  called  an  Ex- 
ponent. 

1.  An  exponent  may  be  integral,  fractional,  positive,  or  negative. 

2.  An  integral  •positive  exponent  of  a  quantity  denotes  a  ro^VEii 
of  that  quantity. 

3.  A  positive  fractional  exponent  of  a  quantity  denotes  a  koot 
of  that  quantity.  Thus,  x^  is  the  same  as  the  fourth  power  of 
ic,  or  X  X  X  X.     But,  x^  is  the  same  as  the  fourth  root  of  x. 

4.  The  fractional    exponent   may  combine   both    a  power    and 

3 

a  root.      Thus,  a;'^  is  the  same  as  the  fourth  root  of  x  cube. 

5.  A  negative  expoiient  of  a  quantity  indicates  that  the  rccij)- 
rocal  of  the  quantity  is  to  be  taken  with  the  sign  of  the  ex- 
ponent changed.  Thus,  x~'^  is  the  same  as  -7  and  — -  is  tlie 
same  as  a; 2.      {Vide  22,  1.)  ^ 

6.  A  letter  may  represent  any  exponent:  as  a:"*,  read  x  wi'* 
power. 

7.  Roots  are  also    expressed,   as   in  Arithmetic,   by  the  signs 

1  J  |/,  -j/,  etc.      Thus,  x^   is   the  same  as  1/a;',  and  av  is  the 

V/ — 
same  as  V  «*  . 

8.  When  no  exponent  is  expressed,  1  is  understood.  Thus, 
a  is  the  same  as  a}. 

9.  Any  quantity  leaving  0  for  an  exponent  is  the  same  as  1. 
Thus,  a°  is  1.     {Vide  6S,  ex.  1.) 

14.  Of  Coefficients.  —  Instead  of  the  expression  a  -\-  a,  we 
may  write  2a;  for  a  -\-  a  -{■  a,  we  may  write  3a/  for  —  x  —  a*, 
we  may  write  —  2x ;  for  —  x  —  x  —  x  —  x  —  x,  we  may  write 
—  ox.  In  each  case  the  figure  standing  before  the  letter  shows 
Jiow  many  times  the  letter  is  taken  addltivcly  or  subtractivcly. 
This  figure  is  called  a  Coefficient. 


12                   DEFINITIONS     AND     EXEECISES.  { 

1.  A  coefficient  may  be  integral,  fractional,  positive,  or  nega-  i 
tive.      Thus,  5x,  Ix,  and  —  ^x. 

2.  A  coefficient  may  be  represented  by  a  letter ;    thus,  bx^. 

3.  When  no  coefficient  is  written,  1  is  understood;  thus,  a  is 
the  same  as  la.  i 

4.  The  expression  Ox  is  the  same  as  0.  | 

15.  Symbols  of  Kelation.  —  The  sign  =   indicates  that  the 
quantities   between  which    it   is   placed   are   equal;    thus,   x  =  i/  \ 
signifies  that  x  equals  y.  [ 

1.  The  whole  expression    of  which  the  sign   =  is  a  part,  is 
called  an  Equation. 

2.  That  part  of  an  equation  on  the  left  of  the  sign  =  is  ] 
called  the  First  Member.  j 

3.  That  part  of  an   equation   on   the  right  of  the  sign  =   is  I 

1 

called  the   Second   Member.      Thus,  2x  -{-  St/  =  a  —  5b  -{-  c   is  ' 

an    equation    of  which    2x  -f  Si/  is   the   first   member,  and  a  —  i 

56  + c  is   the  second  member,   and    the  whole  is  read  thus:    2x  \ 

plus    Sy    equals    a    minus    5b    plus    c;    which    means    that    the  I 

numerical  value  of  the  first  member  is  the  same  as  the  numeri-  j 

cal  value  of  the  second  member, — thus,  3x4r-f2x5=!!0  —  i 

4  +  16,  or,  22  =  22.  ^ 

16.  The  sign  >»  or  <;  indicates  that  the  quantities  between  | 
which  it  is  placed  are  unequal,  the  quantity  on  the  side  of  the  j 
opening  being  the  larger.  Thus,  x'^  y  indicates  that  x  is  ; 
greater  than  y\  also,  x  <^y  indicates  that  x  is  less  than  ?/.  ; 

1.    The  whole   expression    of  which    the  sign    <1  or  ]>  forms 

a  part  is  called  an  Inequation.     Thus,  2x  -\-  5y'^  a  —  b  -\-  2J,  i 
is  an   inequation   of  which  the  first  member  is  greater  than  the 

second.  ; 

IT.    The    Signs   of  I'roportion   are    thus    written,    :    ::    :,    ami  j 

a  :  b  :  :  c  :  d,  is  read  a.  w  to  b,  as  c  /.s  to  d.  \ 


DEFINITIONS     AND     EXERCISES.  13 

18.  The  .sign  OC  indicates  that  one  quantity  varies  as  an- 
other.    Thus,  xozy  signifies  that  x  varies  as  y. 

19.  An  algebraic  expression  is  one  involving  letters  and  signs. 

1.  A  Monomial  is  an  algebraic  expression  consisting  of  one 
term.      Thus,  6x^y. 

2.  A  Binomial  consists  of  two  terms.     Thus,  6x'^y  +  Ahc. 

3.  A  Trinomial  consists  of   three  terms.      Thus,  x  -{■  2y  —  4c. 

4.  A  Polynomial  consists  of  many  terms.  Thus,  x  •{•  Ay  — 
32+  6. 

5.  The  tei^m^  of  a  polynomial  are  separated  by  the  signs 
-j-   or  — . 

6.  A  monomial  is  positive  or  negative  according  as  the  sign  is 
-}-   or  — . 

20.  Sbiilar  terms  are  such  as  have  like  letters  and  exponents. 
Thus,  ^x^y  and  2a^y  are  similar;  but  ^x^y  and  2xy'*  are  dis- 
similar. 

21.  A  polynomial  is  homogeneous  when  the  sum  of  the  ex- 
ponents in  all  the  terms  is  the  same.  Thus,  AaPy*  +  ^^^  — 
x*z-\-iiXymnp  is  homogeneous,  since  the  sum  of  the  exponents 
in  each  term  is  5. 

22.  The  recij^ivcal  of  a  quantity  is  1  divided  by  the  quan- 
tity.    Thus,  -  is  the  reciprocal  of  x. 

1.  The  reciprocal  of  a  /inaction  is  the  fraction  inverted. 
Thus,  the  reciprocal  of  -  is  -. 

23.  The  sign  /,  is  the  same  as  the  words  therefore,  hencCy 
or  consequently. 

24.  The  sign  '/  is  the  same  as  the  word  because. 

25.  The  letters  of  a  term  are  usually  written  alphabetically, 
though  this  order  is  not  essential.     Thus,  ^nhc  is  the  same  as  36co. 


1 4             I)  i:  F 1  X  I T 1  (J  X  xS    A  N  I)    i:  x  i: li  c  i  s e  s .  j 

i 

2S.    The    tei'Ris    of  a    polynomial    are    usually   arranged    with  ', 
reference  to  the  exponent  of  the  leading  letter.      Thus,  x^  -f-  ox^y 

4-  10a::^y  +  lOicy -f  ^^y^-\-y^i  where  x  is  considered   the  lead-  ; 
infi  letter. 

1.  Of  two  polynomials  involving  the  same  letters,  thiit  is  said  ^ 

to  be  algebraically  the  greater  whose  leading  letter  has  the  greater  i 

exponent.     Thus,  x^  —  Sx^y  +  Zxy"^  —  if  is  greater  than  x^  —  ^xy  \ 


2V.  EXAMPLES. 

Involving  tlie  JPrececiiiag  Definitions. 

1.  Convert  into  algebraic  language  the  square  root  of  seven  a 
square,  added  to  five  a  multiplied  by  m. 

Am.    V^ia^  -\-  oam^  or  (7a^  +  5am)^. 

2.  Convert  into  algebraic  language  three  times  the  cube  root 
of  X  square,  diminished  by  the  square  root  of  five  7??,  multiplied 
by  ti  square,  increased  by  twice  the  fifth  root  of  x. 

Am.  3a;3—  {pmn^  +  Ix^y,  or,  ox^  —   i^Smn^  +  2  ^^ 

3.  Convert  into  algebraic  language  the  fifth  root  of  the  sum 
of  X  and  y.  Ans. 

4.  Convert  into  algebraic  language  the  square  root  of  x  in- 
creased by  the  cube  root  of  x  square  and  the  square  root  of  x 
cube.  Ans.  V x  +  V x^  +  Vx^,  or,  x^ -\-  x'^ -{-  x'^ . 

5.  Convert    into    common    language    the    algebraic    expression 

3x'  4-  (2x)^.  ^,^g     S  Three  times  the  fflh  power  of  x  in- 

\  creased   hy  the  square  root  of  two  x. 

6.  Write  in  common  language  the  following  algebraic  expres- 
sions :    o.r"  —   ^Ti^-f  12.T7/,  x  -f   y/.T^  +  —    and 

x^  .^ 

•^  -f    V.r*+    y'.r^  -f  f^~^  .r\ 


DEFINITIONS     AND      EXERCISES.  15 

-y^'  7.     Write    in    common    language    the    following    expressions : 

; p= —   — -  x^  x-\-y 

X1/+   \4x^  —   ^5x^  4-  'dx.,  -j=  7x^  -f  8am., ->  -^/-la  —  bh. 

if  ^       J 

'J^'     8.  Write  in  common   language  the  following   expressions: 

c^c^^a^  —  h',  and  [x  +  Qi  —py^l(ci^  —  5^2^. 
2        ^c 

Kemark. — The  great  advantage  of  algebraic  symbols  has  been 
seen  in  the  previous  examples.  By  them. are  obtained  both  brevity 
and  perspicuity. 

28.  The  NUMERICAL  VALUE  of  an  algebraic  expression  is  the 
value  found  by  arithmetical  reduction  on  affixing  a  numerical  value 
to  each  of  the  letters  composing  the  expression. 

,  EXAMPLES. 

— ^        .  y  - 

1.  What  is  the  numerical  value  of  the  expression  x  -{•  2a^  when 
x=  5,  Ans.    5  +  2x5^,  which  is  55. 

2.  AVhat  is  the  value  of  the  expression  5ab  +  3a^c  —  ??m  w'hen 
a  =  2,  Z;  =  3,  c  =  4,  ?;i  =  5,  and  n  =  6. 

Ans.    5.2.3  +  3.214  —  5.6.  =  48. 

2  3  5 

3.  Find  the  value  of  a^  +  ¥  -\-  c^  when  a  =  8,  &  =  32,  c  =  4, 
(Vide,  13,  4.)  Ans.   8^  +  32^'  +  4^  =  4  +  8  +  32  =  44. 

3  11 

4.  Find  the  value  of  x^  +  5x^  +  ^^  when  x  =  64.     Ans,  534. 

3  ?  1 

5.  Find  the  value  of  (x^  +  2/^  +  ^^)  •  ^>  when  rr  =  4,  ?/  =  8, 
"~"  z  =  16,  772  =  2.  Ans.  28. 

6.  Find  the  value  of  (x  -{-  y)(x  —  y)  when  .t  =  3,  ^  =  3. 

^4ws.  0. 

7.  Find  the  value  of  (x  +  y)  (x  +  ?/)  when  x  =  4,  2/  =  3. 

^ns.  49. 

8.  Find  the  value  of  (x  -{-  y)(ci  -\-  Z>)  (x^  —  ?/^)  when  x  =  4, 
3/  =  4,  a  =  2,  5  =  3.  ^Tzs.  0. 

9.  Find  the  value  of  (x  -\- y  +  Sa  +  2h')(x^  —  7/)  when  x  =  2, 
_y  =  1,  rt  =  4,  ?>  =  3.  ^;?.s.   147. 


16  DEFINITIONS     AND     EXERCISES. 

ahc 

10.  Find  the  value  of  ^^  _|_  ^^  _^  ^^  when  a=  1,  Z>  =  2,  c  =  3. 

1.2.3   Q_ 

^'''*  1.2  +  1.3  +  2.3=  11' 

11.  Find    the   value    of    c^  +  6'  _  ^2   ^j^gn    ^^=50,    h  ==  50, 
c  =  40.  2c  ^^^g^  20. 

12.  Find  the  value  of  x  and  ?/  in  the  equations  x  =  (a  -\-  h^.c, 
and  ?/=(«+  6).c,  c  =  20,  a  =  1,  &  =  2.  « 

h  Ans.  x—QO,y=  30. 

13.  Find  the  value  of  x  and  y  in  the  equations,  x  =  (a  -{-}))  x 
??2cZ  —  (c  4-  (i)  X  nh,    and    ?/  =  (c  +  cQ  X  «?^  —  (a  +  Z>)  X  c??/., 

a(i  —  6c  ad  —  he 

when  a  =  2,  6=1,  m  =  78,  c  =  7,  rf  =  2,  71  =  79. 

Ans,   a;  =  81,  3/  =  72. 

14.  Find  the  value  of  x  in  the  equation  »  =  |  (^  +  ^6^  —  4^), 

if  a  =  4,  6  =  5.  Ans.   x  —  16. 

1     1 

15.  Find  the  value  of  x  and  y  in  the  equations,  rB=    ^^   ^ 


1         1 

w^^  —  n^ 


and  y  = — —  when  m  =  9,  ?i  =  4.  J.?is.   a;  =  6,  ^  =  |. 


71 


16.  Find  x  in  the  equation  a:= (671—  VaW4-62wi2__aV), 

n^ — m^ 

when  cf  =  2,  6  =  3,  wi  =  5,  7i  =  3.  -^tis.    a;  =  1|. 

17.  Find  CK  and  y  in  the  equations  x  =  (^^  —  i^?^)  X  ac  ^^^ 

ac  —  6c 
(am  —  7ic)6a     _,^  <?     x        o  a       ■,       ^ 

y  =  ^ =^— ,    when    a  ^5,    6  =  3,    c  =  4,    cl=6,   m  =  l, 

ad  —  be 

71  =  1. 

18.  Find  x  and  3/   in    the   equations   a;  =  i^_lL__Zf   and  y  =a 

^^±-^,  when  c  =  25,  a  =  2,  6  =  3. 

6  

19.  Find  x  and  2/  in  the  equations  x  =  i(6  -f-  \^  and 
^      36 


y  =  1  (6  —J^g  — 6'^  ^lien  a  =  9,  6  =  3. 


36 


DEFINITIONS     AND     EXERCISES.  17 


cc~^  —  ?/~^ 


Q*4     _     -T/^ 

± —  when  X  =  16, 


IJ     3_x     2 


20.  Find  the  value  of  the  expression  ___L^^,  when  a;  =  2, 

21.  Find  the  value  of  the  expression 

22.  Find  the  value  of  the  expression   locc*+ '^*  + '^  =  + ',  when 
aj  ^  2,  ?/  =  2,  2  =  2,  and  a  =  1,  6  =  1,  c  =  1. 

23.  Find    the  value  of  the  expression  (a  +  xf  +  2x^y,  when 
X  =:  5,  y  =  *J,  a  =  3. 

24.  Find  x  in  the  equation  x  =  IQ)  -^   ^_362+2  ^2(a+6^), 
when  a  =  17,  6  =  3.  ^«s.   x  =  2. 


NOTATION. 


29.  The  fundamental  law  of  numeration  in  Arithmetic  is  this: 
Figures  increase  from  right  toward  the  left  in  a  tenfold  ratio. 

If  we  wish  to  express  a  number  of  more  than  one  figure  by 
means  of  letters  this  law  must  be  observed.     Thus: 

A  number  between  0  and  10  is  expressed  by  any  letter ;  as,  z. 

A  number  between  10  and  100  by  two  letters.     Thus,  10a;  +  y- 

A  number  between  100  and  1,000  by  three  letters.  Thus, 
lOOx  4-  10^  +  z. 

This  may  be  continued  to  any  extent. 

1.  Find  the  value  of  the  expression  \0x -\- y,  when  x=2, 
y  =\  ',  03  =  4,  y  =  o,  etc. 

2.  Find  the  value  of  the  expression  100a;  +  10^  +  z,  when 
x=4,  ?/=0,  2=7,  etc. 

3.  Find  the  value  of  the  expression  xy,  when  x  =  2,  y==l. 

Ans.    2,. 

4.  Find  the  value  of  the  expression  xyz,  when  x  =  4,  y  =  0, 

2  =  7  A?is.    0. 

2 


CHAPTER    II. 

ADDITION — SUBTRACTION — MULTIPLICATION — DIVISION. 

ADDITION. 

29.  Addition,  in  Algebra,  consists  in  finding  the  simplest  ex 
pression  for  the  sum  of  several  given  expressions. 

30.  By  Definition  14,  we  have  a-fa  =  2a,  a -\- a -\- a  =  "da, 
^\2a  -\-  Za  ■=  a  -\-  a  -\-  a  -\-  a  -\-  a  ■=  5a.  Hence,  to  add  similar 
positive  monomials, 

Add  the  coefficients  and  annex  the  common  letters, 

EXAMPLES. 

(1.)  (2.)  (3.)  (4.)  (5.) 

1 

Add  3(X  Qix  Sax  3a^x^  4a'x* 

to  5a  11a;  7ax  5a^x^  5a^x^ 


Ans.  8a  11 X  \Oax  Sa^x^  9alx^ 

6.  Add  together  4.r,  5x,  6x,  9.c  and  25.x.  Ans,    49ic. 

7.  Add  together  la'^x^y,  da^x^i/,  9a\x'*y  and  a^x'^y. 

Ans.  2Wx^i/. 

31.  By  Definition  14,  wq  have  —  a  —  a  =  —  2a,  —  a  —  a  —  a 
=  —  3(2,',  —  2a  —  3a  =  —  a  —  a  —  a  —  a  —  a  =  —  5a.  Ilcncc, 
to  add  similar  negative  monomials, 

Add  the  coefficients,  annex   the   common   letters^;  and  to  the 
result  iirefu:  the  sign  — . 


ADDITION. 

19 

EXAMPLES. 

(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

Add 

—  3a 

—    8a 

—  lOalc^ 

—    5a2x3 

—  Ila3.x* 

to 

—  5a 

—    5a 

—  lOa^x^ 

—  Ua^x^ 

—  20a3x^ 

Arts. 

—  8a 

—  13a 

—  20a'x^ 

—  ISa^ 

—  31ay 

6.  Add    together    —  5a^,    —  3a2,    —  4a2,    —  Ta^,    —  10a-    and 
—  a\  Ans.  —  30a2. 

7.  Add  togetlier  —  Sx^y,  —  ^x^y,  —  Tx^y,  —  Qx^y  and  —  x^y. 

Ans.  —  24x2y. 
32.  When  the  monomials  are  not  similar, 

Write  the  terms  after  each  other^  retaininrj  the  given  signs. 


EXAIMPLES. 

(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

Add 

a 

3a 

2a2 

5x3 

1 
—  4x^ 

to 

h 

2x 

3a 

—  3x2 

3x» 

Ans. 

a-j-h 

3a4-2x 

2a2H-3a 

5x^  —  3x2 

—  4x3+3x' 

6.  Add  together  a,  —  b,  c  and  —  x,  Ans.  a  —  &  -f  «  —  ^• 

33.  To  add  similar  monomials  with  unlike  signs, 
Find  the  sum  of  the  positive  monomials  ly  30. 
Find  the  sum  of  the  negative  monomials  hy  31. 
Take  the  smaller  coefficient  from  the  larger  and  annex  the 

common  letters. 
Prefix  the  sign  of  the  larger  coefficient  to  the  result. 


EXAMPLES. 

(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

Add 

oa 

4x2 

-7x 

—  20x* 

9x5 

to 

—  2a 

-      X2 

3x 

llx^ 

—  12x5 

Ans. 

S'z 

ox^ 

-  4x 

—    9x^ 

—    3x5 

20 


ADDITION. 

Add 

together 

(6.) 

(7.) 

(8.) 

(9.) 

(10.) 

5a 

5x1/ 

3x2 

2x^ 

Axyz 

—    Sa 

—  lOa^y 

5x' 

—    5x^ 

5xyz 

7a 

—  13xy 

— 

4a:= 

—    2>x^ 

—    dxyz 

Sa 

8X2/ 

IQx^ 

-    7a3 

Zxyz 

12a 

l^xy 

— 

Vlx"" 

—  10^3 

—    Ixyz 

—  13a 

—  20xy 

—  IQixy 

- 

la? 

sJ' 

—  X^xyz 

,      16a 

^j? 

—  ISic^ 

—  XAiXyz 

Ans. 

11.  Add  together  a\v,  da'^x,  — 7a^.T,  Ida^x,  — da^x,  — 200^0;, 
and  —  3a^x,  Ans.    —  20a'a7. 

12.  Add  together  5,  —  4,  17,  —  30,  and  80.  Ans.    68. 

13.  Add  together  Ix^y^,  —  9x^i^^,  Ax'^f,  —  x'^y'^.  Ans.  x^y\ 

14.  Add  together  Axy'^^  7.t?/^,   9xy^,   —  11  xy"^.  Ans.    Sxy*. 

15.  Add  together  15x^y,   IGx^y,   18x^y,   —  Six^y. 

16.  Add  together  —  12a:y,   —  20xy,  and  30xy. 

17.  Add  together  —  19xy,   —  21a:2^^  ^^j^^j  Aoxy, 

18.  Add  together  —  30.xy,  AOxY,  and  l^xy. 

19.  Add  together  Ax,  5x,  —  3x,  and  —  6x,  Ans.    0, 

20.  Add  together  5xyz,  Ixyz,  and  —  12xyz. 

34.    To  add  polynomials  having  in  each  similar  terms, 

Arrange  the  iwJynomiaU  so  that  similar  terms  stand  under 

each  other. 
Add  each  column  of  terins  hy  33. 


EXAMPLES. 

(1.) 

(2.) 

(3.) 

Add 

x-\-    y 

3x  +2y  +    z 

Ix  —    5y  4-6^—10 

to 

x-2y 

5x  —  Ay  —  Qz 

Ax  +  10//  —  9^4-  30 

2x—    y 

Sx  —  2y  —  5:: 

Ux -\-     ov  — 8.:+20 

ADDITION.  21 

(4.)                                (5.)  (6.) 

Add      4x-\-3t/—2z           3x^+2?/'— 42^+10  ic^+  x'^^  5x}j 

— 5ic+4z/+62!           4x2— 2^/3+02"— 10  3a;^— 2x2^+   Trr^/ 

"jx—Sij—^z       —5x'^-\-3i/—2z'^+\6  — 90)^—4x2^— 13.T^ 

4x—  2/+   z           6.^2—8/ +42"— 16  7x'^+Sx^y-\-     xy 


Am.    10x—2y—4:Z  Sx^—5f-{-3z^—  1  2x^—2x2^ 

7.  Add  together  the  polynomials  3a  +  25  —  5c  +  12x  —  10, 
—  7a  4-  36  —  Cc  —  13x  +  12,  4a  +  36  —  10c  —  5x  +  8,  and 
10a  —  76  +  3c  — x+  13. 

Sohitio7i. 

Sa-\-  2h  —    5c  +  12x—  10 

__7a  +  36—    6c— 13ic+12 

4a  +  36  —  10c  —    5x  +    8 

10a  —  76  +    3c  —      x-j-  13 

10a  +    &  — 18c—    7x  +  23 

8.  Add  together  the  polynomials  7a?x  +  56'  —  7jn^  +  14ri, 
da^x  —  7n-i-  96'  —  5??i2^   __  106'  +  Am^  +  8?i  —  lOa'x. 

Solution, 
7c?x  +    56'  —  7m5  +1471 
3a=^x  +     96'  —  5m2  —    77t 
—  lOa^x  —  106'  +  4m2  +     ^n 


46'  —  8m2  +  15?i 

9.  Add  the  polynomials  2x  +  3^  —  z  and  2x  —  3?/  +  2?. 

Ans.    4x. 

10.  Add  together  x  +  2^  +  c  and   —  x  +  2?/  —  c. 

-472S.     4^. 

11.  Add  together  x  +  Sy  —  5^  +  ^  +  9,  6^  —  2x  +  32  —  1  —  5^, 
7/7*  +  1  —  3z  -\-  y  —  X,  —  g  —  8  —  3^  —  x,  and  3  +  52  —  9?/  — 
g  +  Ix.  Ans.    Ax  +  3y  +  5g  +  4. 


22  ADDITION. 


12.  Add  together  Zx'^y  +  2a;y  —  Wij  +  oxij\  \x^i)  —  Zxy^  — 


1 


"Ixhf  +  ^rc^y,  —  X^xhj  +  8xj/'  +  5xy  +  7x2y  and  lOx^  —  '^W 
1  1 

i    1 

13.  Add  together  2xy  +  "Ix^y  —  Sa;^^  +  Zx  ^y%  8xy^  —  5xy^  + 

2xy^  —  ox^^y^^,  15x^y^  —  18x^y^  —  ox^y  +  7xy  and   —  2xy  — 

1    1 
"jx^y  +  4a;y  —  Sx^y.  Ans.  llxy  —  llx^y  —  3xy^  —  3x^y^. 

14.  Add  together  dx^y^  —  2xyz  +  ox,  Ixyz  —  ox  —  5x^y^  and 

—  5xyz  +  2x^yK  Ans.  0. 

15.  Add  together  x^'  -\-  x^  +  re*  —  x\  ox^  +  7x^  —  8a5*  + 
5a;^  and  •—  2x^  +  S.-c^  +  7a;*  —  lOx^      ^«s.  9a;^  +  6x^  —  6a;^. 

16.  Add  together  oxy  +  2a2Z>2  _  5^2?^  +  40,  3mn  —  20  —  5rt' 

?>=*  +  2a:y  and  2mu  —  10  +  Sa^t^  _  7^;^. 

3  5 

17.  Add  together  x"^  -{-  y^  -\-  xy  -\-  x^y^  -\-  x,  —  7x  —  4x^y^  — 

3xy  —  Sy^  +  2x-  and  4.xy  —  tx^y^  -\-  ox  —  5y^  +  3x^. 

18.  Add  together  3x*  —  c  +  7m  —  n  -\-  10,  --  5^"  —  2c  --  7m 
+  ^  —  15,  7cc^  —  2c  +  3??i  —  2?i  and  —  lOx''  —  5c  +  6m  -\-  3x 

—  3c  +  2x\ 

19.  Add  together  Ax^y  -f-  5.x  4- 3m  +  3/  +  ^2^1  'h  ^^i  Zx^y  — 
lOx  -f  4m  —  ^y  -\-  ^p(l  —  82  and  —  7m  +  7?/  —  7pq  -\-  6z  -{■  Qx 

—  7x^y, 

20.  Add  together  x^  +  S.t:^?^  +  5?/2  +  4^  +  32;^  —  lOrc^m  — 
7y^  +  5;)  —  10^2  +  5x2  j^jj^  g^a  _  9^,  _|_  3^  _^  7^2^  __  5^2^ 

21.  Add  together  3x^  +  2xy  +  ?/2,  —  2xy  +  Sj/^  +  Sx'  and  — 
Ay"^  +  4a:;y  +  2a:^ 

22.  Add  together  x^  -f  2xy  4-  ^'  and  x^  —  2xy  +  ?/^ 

23.  Add   together   x?  +  Sx^?/  4-  3x7/'  4-  y^   and   x?  —  ox^y  -\- 
oxy"^  —  2/'. 

24.  Add  together   x*  4-  ^^^y  +  Gx-^j/^  +  4x7/'  +  ?/^  and  x"*  — 
A'3i?y  -\-  6xy  —  4x?/'  +  ?/*. 

25.  Add  together  x?  -|-  x?/  -\-  y"^  and  x-  —  xy  -\-  _y'. 


ADDITION.  23 

2G.  Find  the  nuiuerical  value  of  the  last  five  examples  when 
a;=2  and  ^=2.  Ans.  2P'=  5G,  22"^^=  16,  23''^=Gi,  24'"  = 
256,  25''^=  16. 

^  11  1      L 

2/.  Add  together  cc  -j-  x^y^  +  2/  ^^id  re  —  x^y''^  +  2/- 

28.  Add  together  x^  +  Gx'y  +  lorcy  +  20.^^+  lox'^y* +Gxy^ 
+  3/^  and  x^  —  6x*y  +  15a:''_?/^  —  20x^j/^  +  lo.x^  —  6xy^  +  2/^' 

29.  Find  the  numerical  value  of  the  last  two  examples,  when 
05=1,  2/ =  2.  Ans.  6  and  730. 

30.  Add  together  3  (x  +  y),  2  (x  +  y)   and  8  (x  +  y). 

Ajis.  13(x  -\-  y). 

31.  Add  together  2  (x^  +  7/)  +  5  (x -}-  y  -\- z)  +  4:  (x^  +  2y^ 
and  6  (aj'  +  y')  —  4(a;+y  +  ^)  —  2  (cc^  +  2?/2),  where  rc=l, 
y  =  2,  ;s  =  3. 

^7w.  8  (x2  +  2/0  +  G^  +  2/  +  ^)  +  2  (x'  +  2y»)  =  96. 

32.  Add  together  x  +  I  (x  -\-  y  +  z)  -{-  4:,  y  +  m  (x  +  y  -\-  z) 
—  3  and  z  -\-  n  (x  -\-  y  -j-  z)  +  5,  and  find  the  numerical  "value 
when  x=  1,  y  =  2,  s  =  3,  Z  =  4,  ?n  =  o,  n  =  6, 

Ans.  x  +  y  +  z-i-(l+m-{-n)(x  +  y-\-z')  +  6=  102. 

33.  Add  together  3x  (1  +  2y')  +  9,  5x  (1  +  2^/)  —  7,  —2ix 
(1  ^  2?/)  +  12  and  7ix  (1  +  2y)  —  8.     ^?zs.  13a;  (1  +  2y')  +  6, 

34.  Add  together  7  +  4-  (2c  +  d  —  m)  +  3a;2,  _  8  +  K'^c  + 
d  —  77i)  —  5x^  and  8  —  -^  (2c  -\-  d  —  7n)  +  2x^ 

^/?s.  7  +  I  (2c  +  c^  — m)- 

35.  Add  together  0  (x  +  y)^  +7,  4  (a;  +  t/)^  —  3  and  —  8 
(a;  +  y)*  — 4.  ^«5.  (a;  +  y)^ 

36.  Add  together  x  ■\-  y  and  x  —  y.  Ans.  2x. 

35.  By  the  last  example  we  see  that 

The  sum  of  two  numbers  added  to  their  difference  gives  twice 
the  lar-ger  number. 


24  SUBTRACIION. 


EXAMPLES. 


1.  (12  +  5)  +  0'2  —  5)  =  2  X  12  =  24. 

2.  (41-  4-  3)  +  (4i  -  3)  =  2  X  4^  =    9. 

3.  (6  +  20  +  (6  -  2i)  =  12. 

4.  (17-30  +  (17  +  30  =34. 

5.  (8i  -  20  +  (Hi  +  20  =  17. 

6.  (3i+10+  (3k-  10  =6i. 

7.  (4  -  2)  +  (4  +  2)  = 

8.  (8x  +  2?/)  +  (8x  —  2?/)  =  16*«. 

9.  (2ix  +  Siy)  +  (2ix  -  3^2/)  = 

10.  (5x*  +  2y)  4-  (5x^  —  27/)  = 

11.  What  is  the  value  of  the  last  three  examples  when  x  =  5, 

7/  =  3  ?  Ans.  8"^  =  80,  9^'*  =  25,  10'^  =  6250. 


SUBTEACTION. 


36.  Subtraction  in  Algebra  consists  in  finding  the  simplest 
expression  for  the  difference  of  two  given  expressions^  or,  in  finding 
what  quantity  added  to  the  subtrahend  will  produce  the  minuend, 

37.  Of  the  two  given  expressions  that  which  is  to  be  subtracted 

is  called  the  subtrahend;    the  other  is  called  the  minuend. 

3S.  To  find  the  difference  of  two  similar  monomials : 
This  difference  may  always   be  expressed    by  either   a  positive 
or  a  negative  quantity,  each  result  depending  upon  which  of  the 
given  expressions  is   taken  for  the  minuend  ;   thus, 

Vide  24,  36. 

1.  From   5a  subtract  3a  and  we  have  2a  •.*  3a  +  2a  =  5a. 

2.  From  3a  subtract  6a  and  we  have  —  2a  •/  5a  +  ( —  2(r)  =  3a. 


SUBTEACTION.  25 

3.  From  5a  subtract  —  3a  and  wc  have  8a  *.•  —  3a  -f-  8a  =  5a. 

4.  From  —  3a  subtract  5a  and  we  have  —  8a  *.•  5a  -f  ( —  Sa) 

=  —  3a. 

5.  From  —  5a  subtract  3a  and  we  have  —  8a  *.•  Sa  +  ( —  8a) 

=  —  5a. 
G.  From  3a  subtract  —  5a  and  ^xe  have  8a  •.•   —  5a  -f  8a  =  3a. 

'7.  From  —  5a  subtract  —  3a  and  we  have  —  2a   *.•    —  3a  -j- 

( —  oa)  =  —  8a. 
8.  From  —  3a  subtract  —  5a  and  we  have  2a  •.•   —  oa  -f  2a 

=  —  3a. 

These  may  be  arranged  as  follows : 

(1.)     (2.)       (3.)         (4.)         (5.)       (6.)  (7.)         (8.) 


From 

f 
oa 

oa 

5a 

—  oa 

—  oa 

3a 

—  oa 

—  3a 

take 

3a 

2a    • 

5a 

—  3a 

5  a 

3a 

—  5a 

—  3a 

—  5  a 

Ans. 

-2a 

8a 

—  Sa 

—  8a 

8a 

—  2a 

2a 

By  comparing  (1)  and  (2)  it  will  be  seen  that  the  difference 
between  5a  and  3a  is  expressed  either  by  2a  or  by  —  2a. 

There  is  a  similar  relation  between  (3)  and  (4),  (o)  and  (6), 
(7)  and   (8). 

Each  of  these  results  may  be  obtained  by  the  following  rule : 

Consider  tJie  sign  of  the  subtrahend  changed. 

If  the  signs  are  then  alike,  add  the  coefficients,  prefix  the  com- 
mon sign,  and  amiex  the  common  letters. 

If  the  signs  arejinlike,  subtract  the  less  coefficient  from  the  greater, 
prefix  the  sign  of  tJie  larger  coefficient,  and  annex  the  common 
letters, 

EXAMPLES. 

(1.)     (2.)       (3.)  (4.)         (5.)       (G.)         (7.)        (8.) 

From  lOx      \Qx    —  lOx     —lOx         2>x    —    3.7;  Zx    —    ox 

take     3x  —  3x  3x      —    ox       lOx         10.x  —10.x    —  lOrc 


Ans.     7x      lox     —  13.T     —    7x    —7x—\3x        lo.x  Ix 

o 
O 


SUBTRACTION. 


(9.)  (10.)  (11.)  (12.) 

1      3 

From  oci^j^  Iba^x^y  \2axyz  —  Ax^ 

1      3 

take  "Ici^x  —  IQa^x'^ij  30ax7/z  7x^ 


1       3 


Ans.            — 2a^.»                 ola~x^f/            — 18axf/z  — 11. ^;*      \ 

13.  From  7a\c  subtract  ^ci'x.  Ans.               ] 

13                                             13  ! 

14.  From  r-  IGa^x'^T/  subtract  loa^x'^i/.  Ans.               ' 

15.  From  SOaxi/z  subtract  —  12axi/z.  Ans.  i 
IG.  From  7x^  subtract  —  4x*.  Ans.  ^ 
17.  From  lOx^  subtract  lOa;^  Ans.  0. 

39.  To  find  the  difference  of  two  monomials  not  similar:             ; 

! 

Change  the  sign  of  the  subtrahend  and  ivrite  it  after  the      i 

minuend.  j 

EXAMPLES.  j 

(1.)         (2.)          (3.)            (4.)            (5.)  (6.)             I 

From    a             5  a             3.^              7?/'              4:xy  —  12x^ 

take    Z)       — 2Z)             Ay              Am               3x  — 3a; 


Ans.  a  —  h     ba-\-2h    3x  —  Ay     7y^ — Am    Axy  —  3x     — 12x'+3a; 


7.  From  3a;  subtract  2y. 

8.  From  7a;'  subtract  2a;^. 

9.  From  10a;  subtract  —  Am. 


10.  From  5a;?/  subtract  12. 

11.  From  12  subtract  — 4a;. 

12.  From  15  subtract  — 2a;^ 


40.  To  find  the  difference  of  two  polynomials : 

Write  the  j^ohjnomial  taken  as  the  subtrahend  under  that 
tahen  as  the  minuend^  j;/«c?Vi^  similar  terms  under  each 
other  and  those  not  similar  in  any  order. 

Subtract  similar  terms  by  the  rule  in  38. 

Subtract  terms  not  similar  by  39. 


SUBTRACTION.  27 


EXAMPLES. 

(1.)  (2.)  (3.)  (-t.) 

1  1 


1  1 

From        3a;  —  y         x  —  y         x^  -{-  2x>/  -j-  y^         x  -\-  x^y^-\-  y 
take        2x  —  h         x  -{-  2y       x^  —  2xy  -{■  y^         x  —  x^^y^-{-  y 

,  y  11 

A71S.     X  —  y -\-  h         — 3?/  Axy  2x''^y^ 

5.     From         3a  +  26  —  3c  +  %  —  om  +  2;i  —  7x  —  y-\-10 
take  a  —  36  -f  4c  —  2g  -\-  5m  —    n  -{■  Ax  —  y  —  40 

Ans,  2a  +  56  — 7c  +  6g  —  10m  +  3n  —  llx        +  50 

(6.)  (T.)  (8.) 

From  x^  -\-  oxy  -\-  y^  x"^  -\-  ^U  -\-  y"^  ^ 

take         —  x^y  -\-  3xy  +  xy^  —  x^  -{■  xy  —  y^  x  —  y 

Ans.        x^  +  x^y  +  2/^  —  ^I/^  2a;^     4"      ^i/^  y 

9.  From  x  —  y  subtract  x  —  2y.  Ans.  y. 

10.  From  x  -{■  y  subtract  x.  Ans,  y. 

11.  From  x"^  —  xy  +  y"^  subtract  x"^  —  xy  —  y"^.  Ans.  2y^ 

12.  From  x  -\-    ^xy  -f  y  subtract  x  —  ^xy  -f  y.      Ans,  2  ^xy. 

13.  From  x^  +  ijx^y  +  ^xy^  +  y^  subtract  x^  —  Sx'^y  +  3xy^ — y^. 

Ans.  6x^y  -f-  2_y^ 

14.  From  x'^  -f  4x^y  +  Gx^y^  +  4^xy^  +  y^  subtract  x^  —  4x^y  + 

6x?y^  —  4xy^  -\-  y^. 

15.  From      x^  +  G.^^y  +  ISx''^^  ^  20a:^7/3  +  l^x^y^  +  Ga:^^  +  y\ 
subtract    x^  —  Qx^y  +  15x^7/=  —  20a:^3/3  +  15^2  t/"  —  Ga^y  +  ?/^ 


,5 


Ans.  12x'y  -}- 40x' y'  -^  12xy' 

16.  From   3m  +  2;i  —  ^xy  +  4^)  —  (2  +  2^^  subtract   2m  —  3/1  — 

4p  +  5ary  -|-  2a;  —  q. 

17.  From  ^^3/^  +  ^'^  +  *^^^^  subtract  a;^^^  —  xy  -^  x^y^. 

IS.  From  x^  +  3^:^^  +  3a-2^^  +  y^  subtract  —  x'y  —  3x'y^  +  3x2 

19.  From   a;^  -f  3x'y^  +  3.r2^^  +  y'  subtract  x'y  +  3.r^j/2  _  3,-^2^ 
+  a;*'. 


28  SUB  T  R  A  C  T  I  0  N  . 

20.  From  x  -\-  2tj  —  -1  subtract  x  —  3y  -j-  7. 

21.  From  4.^:  +  7/y  —  G  subtract  2x  +  7?/  +  12. 

22.  From  x  ^  -f  5x  ^  —  7a;  2  subtract  2x  ^  —  Sx^. 

23.  From  1  -{'  x  -\-  x"^  -{-  x^  subtract  1  —  x  -\-  x^  —  x^. 

24.  From  1  +  2x  +  Sx^  -j-  bx^  subtract  1  —  2a;  +  Sx^  —  5x». 

25.  From  1  -[-  5x  -{-  G.x^  subtract  1  —  5x  -J-  Gx^ 

26.  From  x  -\~  x^  -\-  x^  subtract  x  —  x^  —  .x^ 

27.  From  x^  -\-  2xy  -\-  if  subtract  x?  -\-  xy. 

28.  From  o^  -\-  y^  subtract  x^  -j-  x'^y. 

29.  From  x^  —  ^x'^y'^  -|-  Sx^y  —  y^  subtract  x^  —  ?»x/'y  -j-  Zx^y^  — 

x^y\ 

41.  From  the  sum  of  two  or  more  quantities  to  subtract  any 
number  of  quantities: 

Cliange  the  signs  of  the  subtrahends  and  add  the  columns  as  in  33. 

EXAMPLES. 

1.  From  the  sum  of  4x  -\-3y  —  2z  and   —  5a;  -{-  4:y  -\-  Qz  take 

—  7a;  -j-  Sy  -J-  9;3  and  —  Ax -{- y  —  z.     (  Vide  34,  ex.  4.) 

Ans.  lOo;  —  2y  —  4z. 

2.  From  the  sum  of  Sa;^ -f  2^  —  42"+ 10  and  ix^  —  2y^ -\- bz* 

—  10  take  5a;2  —  3y'  -]-  2z*  —  15  and  —  Gx^  -j-  Sy^  —  4z'  -f  16. 

Ans.  8a;2  —  5/ 4- 3;s-' —  1. 

3.  From  the  sum  of  Sx  —  4x7/  -|-  83/^  and  6xy  —  7x  -|-  4y^  -]-  8 
take  3x  +  2xy  —  by^  +  4  and  5x  +  2xy  +  lOif  —  15. 

4.  From  x^  -\-  2xy  -\-  y^,  x^  —  2xy  -j-  y^  and  x^  -\- ^^  -\-  5?/^  take 
a:2  _  5.ry  _  23/=,  2x2  _|_  4,^,^  _j_  9^2  ^^^  _  ^2  _|_  2;cy, 

5.  From  2  (x  —  2/)  +  -5  ^  (x  _  7/)  -f  2:;  and  b(x  —  y)—z  take 
4(.^-2/)  +  ^-2. 

G.   From   3(x  — 7/)-2-f  4   and   2(x  — 7/)-2-f  6   take  4  (x — 
2/)~2-l-8  and  find  the  vakie  when  x  =  8,  y  =  3. 

.4«5.  to  last  jioint  2^. 


S  U  B  T  i:  A  C  T  I  O  N  .  29 

7.  From  7  (.x  +  ?/)^-f  15   and  8  (.x  +  ^)2  —  IG   tul;e  12  (.t  + 
y)2 — 1   and  find  the  value  when  x=lG,  y  =  9. 

-<4;i5.  ^0  /(cwi  />o?V<^  15. 

8.  From   8  (x^ -|- t/^j  2  _|_  2:ry   and   4  (x^ -|- t/^)*^  _j_  S.ry   take   10 

1 
(rc^  -|-  if)  ^  -\-  \xy  and  find  the  vahae  when  re  =  4,   ?/  =  3. 

^l«s.  22. 
42.  In  the  expressions 

+  (+«),  +  (-  «),  -  (+  «)  and   -  (-  «) 

the  sign  before  the  parenthesis  is  called  the  sign  of  operation. 
The  sign  before  the  letter  is  called  the  sign  of  the  quantitij. 
Thus, 

(1.)   -f-  (-}-  <^)  means  that  the  positive  quantity  a  is  to  be  added. 

(2.)   -j-  ( — a)  means  that  the  negative  quantity  — a  is  to  be  added. 

(3.)    —  (-}-  a)  means  tliat  tlie  positive  quantity  a  is  to  be  subtracted. 

(4.)   — ( — a)  means  that  tlie  negative  quantity  — a  is  to  be  sub- 
tracted. 
]5y  performing  tlie  operations  indicated  by  the  sign  of  operation, 

we  have, 

(1.)    +  (+  a)  =  +  a,  (3.)    -  (-1-  a)  =  -  a, 

(2.)   +  (_  a)  =  -  a,  (4.)   _(_«)  =  +  a, 

where  the  sign  in  the  second  member  of  each  equation  is  called 

the  essential  sign. 

(5.)  J5y  comparing  (1)  and  (4)  it  is  seen  that  the  addition  of  a 
positive  quantity  is  the  same  as  the  subtraction  of  an  equal  neg- 
ative quantity  ;  that  is  -\-  (-f-  a)  =  —  ( —  a). 

(6.)  By  comparing  (2)  and  (3)  it  is  seen  that  the  addition  of 
a  negative  quantity  is  the  same  as  the  subtraction  of  an  equal 
positive  quantity  ;  that  is  -[~  ( —  «)  =  —  (-|-  «). 

(7.)   It  is  plain,   tlicn,   that    —  ( —  ah)  =  -f  ah. 


30  S  U  B  T  K  A  C  T  I  0  N  . 

EXAMPLES. 

1.  What  is  the  value  of  3x — (^-\-^x)"l  Ans.  — 2x, 

2.  AVhat  is  the  value  of  3x  -[-  (—  5.x)  ?  Ans.  —  2x, 

3.  What  is  the  value  of  3.x  -]-  (-|-  5x)  ?  Ans.  8.x. 

4.  What  is  the  value  of  3x  —  ( —  5a;)  ?  Ans.  8.x. 

43.  The  subtraction  of  a  polynomial  is  indicated  by  inclosing 
it  in  a  parenthesis  and  prefixing  the  sign  — .  Thus,  x  -{-  y  — 
(x  —  y)  signifies  that  x  —  y  is  to  be  subtracted  from  x  -\-  y. 

Performing  the  operations  we  have  x-\^  y  —  (x  —  y^-=.x-\-  y 
—  X  -{-?/=  2y.  Hence,  to  remove  a  parenthesis  having  a  nega- 
tive sign  of  oj^teration, 

Change  all  the  signs  in  the  parenthesis  and  unite  the  terms  as  in 
addition, 

EXAMPLES. 

1.  Eemove  the  parenthesis  from  a  —  (h -{- c),       Ans.  a  —  h  —  c. 

2.  Remove  the  parenthesis  from  a — (h  —  c).      Ans.  a  —  h -{- c, 

3.  Remove  the  parenthesis  from  a —  ( —  h  -\-  c).     Ans.  a-\-h  —  c, 

4.  Remove  the  parenthesis  from  a —  ( —  h  —  c).     Ans.  a-\-h  -{-  c, 

5.  Remove  the   parenthesis  from    x  -\- 2y  —  4  —  (x  —  3?/  -j-  7). 

(HcZe  40,  ex.  20).  Ans.  5?/  — 11. 

6.  Remove  the  parentheses  from  a  —  \h  —  (c  —  rf)-f-x]. 

Ans.  a  —  (h  —  c  -\-  d  -\-  x')  =  a  —  h  -\~  c  —  d  —  x. 

7.  Remove  the  parentheses  from  a  —  [\h  —  [c — (d — e) — /] — g]], 

Ans.  a  —  h  -\-  c  —  d.-\-e  — f-\-  g. 

44.  By  reversing  the  operations  of  43,  polynomials  may  be 
written  in  various  ways.     Thus, 

1.  X  — 5.x2-{-6.x3-j-7.x*  — 8x^ 

is  the  same  as  x  —  (5.x^  —  G.x^  —  7.x^  -|-  8.x^), 

which   is  the  same  as     x  —  [5x^  —  (Crx^  -f  7.x'')  -f  8x*]. 

AVhat  is  the  value  of  cither  polynomial  when  x=2? 

Ans.  —-114. 


SUBTK ACTION.  ol 

2.  Find  the  value  of  a  —  \[—h^lc—{—d—f)-\-(j']  —  h]] 
when  a  =  1,  h  =  2,  c  =  3,  c^  =  4,  /=  5,  ^  =  G  and  h  =  7. 

45.  Since  x^  y  —  (x  —  ?/)  =  2?/,  it  is  plain  that 

Tlie  difference  of  two  numbers  taken  from  their  sum   (jives 
twice  the  smaller  number, 

EXAJn'LES. 

1.  (12  +  5)  -  (12  -  5)  =  2  X  5  =  10. 

2.  (4i  -I-  3)  -  (4i  -  3)  =  2  X  3  =  6. 

3.  (6  +  2i)-(6-2i)=  5. 

4.  (4  +  2)-(4-2)  = 

5.  (17  +  30  -  (17  -  31)  =  7. 

6.  (7 +  20 -(7 -20=  5. 

7.  (101  +  5|)-(m-5|)=  IH. 

8.  (6|  +  2^)-(6|-2A)=  41. 

9.  (3x^  +  57/)  -  (3x''  -  57/)  = 
10.  (2x5  _j_  3^1)  _  (2x^  —  3/3)  = 

What  is   the  value  of  9'-''^  and   lO'^^  examples   when  x  =  256, 
2/  =  8? 

46.  PROBLEMS    IX    SUBTRACTION. 

1.  If  A  is  worth   5a  dollars  and  15  3a  dollars,  what   is   the 

difference  between  their  pecuniary  conditions? 

Ans.  -\-  2a  or  —  2a  dollars. 

2.  If  A  is  worth  5a-  dollars  and  B  is  in  debt  3a  dollars,  what 

is  the  difference  between  their  pecuniary  conditions? 

Ans.   -{-  Sa  or  —  Sa  dollars. 

3.  If  A  is  in  debt  5a  dollars,  and  B  is  worth  3a  dollars,  what 

is  the  difference  between  their  pecuniary  conditions? 

Ans.  —  8a  or  -|-  8a  dollars. 

4.  If  A  is  in  debt  5a  dollars,  and  B  is  also  in  debt  3a  dollars, 
what  is  the  difference  between  their  pecuniary  condition  ? 

Ans.   —  2a  or  -p  2a  dollars. 

If  a  =  ^3000  what  are   the  answers  of  each  example  ? 


32  M  U  L  T  I  P  L  I  C  A  T  I  C  N  . 

4'1'.  By  distinguishing  what  each  one  is  worth  bj  -\-  and 
what  each  one  is  in  debt  by  — ,  the  connection  of  these  prob- 
lems with  the  previous  principles  is  evident,  (^Vide  3S.).  It  is 
seen  that  the  difference  between  two  quantities  can  be  expressed 
as  well  by  a  negative  as  by  a  positive  result.  In  Arithmetic  it 
is  not  necessary  to  recognize  this  fact,  but  in  Algebra  it  is  of 
the  utmost  importance  to  have  a  correct  apprehension  of  it. 

4§.  Merely  to  find  a  difference  it  is  of  no  consequence  which 
of  two  given  quantities  we  call  the  minuend  or  which  the  sub- 
trahend. After  having,  however,  assumed  one  of  the  quantities 
to  be  the  minuend,  the  result  must  always  he  referred  to  it, 

49.  The  difference  between  7  and  4  is  -j-  3  or  —  3 ;  thus, 


7"^ 

-}-  3  showing  that  7  is 

!  3  greater  than  4, 

+  3j 


^1 

^  »  — 3  showing  that  4  is 

I  3  less  than  7. 

_3j 


50.  The  negative  sign,  then,  serves  to  indicate  some  peculiar 
circumstance  connected  with  the  quantity  before  which  it  is 
placed. 

51.  In  a  given  problem,  negative  quantities  have  a  sense  con- 
trary to  that  which  limits  iiositive  quantities. 


MULTIPLICATION. 


—  52.  Multiplication  is  the  operation  of  finding  the  product  of 
two  quantities. 

—  53.   The  multiplicand  is  the  quantity  to  be  multiplied. 

^    54.  The  luultiplier  is  the  quantity  by  which   to  multiply. 


MULTIPLICATION.  33 

55.  The  multiplicand  and  the  multiplier  may  be  inleichangcd 
at  pleasure. 

^   56.  To  multiply  positive  monomials;   (^Vide  Dcf.  19,  6.): 
By  Def.  10  we  have  x  X  y  =  xy. 
By  Def.  13  we  have  x.x  =  x^  and  x.x.x  =  x^  ,  * ,  x^  X  x^ 

=  xxxxx  =  x^. 
By  Def.  14  we  have  5a  x  3  =  5a  -J-  5a  -|-  5a  =  15a  .  * ,  bax^ 

X  3x^y  =  Ibax^y. 
Hence,   (j:ide  25,) 

Multijjly  the  coefficients  and  add  iJie  exjioncnis  of  like  letters. 

EXAMrLES. 

(1.)  (2.)  (3.)  (4.)^  (5.)^ 

Multiply  7a26  2a''x  IQa^V  lOa^^'  tJ/ 

by  5ax  5a3.x2  2ah  2ah^  3x{f 


Ans.             '6ba'bx  lOa'x'  d'Za'b'  20ah  2\xy 

(G.)^               (7.)  (8.)^  (9.)  (10.) 

Multiply          Ix^y^  ^xyz-'^  Sx'y~^  hx^y^  3a;^?/-* 

t     2  13  11,                           1 

by                \x-y^  l^~  y^*  7x^y^  bx^y^  bx'y* 


Ans.  X  y  ly"^  lbx~y    ' 

.^  5t.  To  multiply  two  monomials  with  unlike  signs : 

By  Def.  14  we  have  — 5a  x  3  =  — 5a  —  5a  —  5a  =  —  15a. 

But  —  5a  X  3  is  the  same  as  3  X  —  5a.     ( Vide  55.) 
Hence, 

Multiply  as  in  56,  and  prejix  tlie  sign  —  to  the  jyroduct. 

EXAMPLES. 

(1.)  (2.)  (3.)  (4.)  (5.) 

It  3_    4  1     ,  11 

Multiply         4a^6        — 2x'^y'^        — 4:m^n'^         — ix'y^*  27n^x'^ 

by        — 2a^b  Sx^f  2fnhS  5.-cV°     ~2n^'x^ 

Ans.      —8a'b^       —<>.'•?/  —Sm^/i" 


(8.) 

(9.) 

^x'yh 

3xyz 

—  Sax^y 

—  2ahc 

34  MULTIPLICATION. 

(6.)  (7.) 

Multiply      — 2x-^y-'^z-^  — Ax-^y-^z-^ 
by                \^  y  ^  Ga^y^^ 

Ans,  —  1 

58.  To  multiply  negative  monomials: 

It  is  plain  that  —  a  s=  —  1  (-j-  a)  and  —  t  =  -j-  1  ( —  h^, 
Plence 

Multiply  —  a  =  —  1  (_[_  a)  Vide  Def.  14,  3. 

by        -6=  +l(-6) 

and  we  have  —  1  ( —  alj^  =  —  ( —  ah')  =  ah.  ") 

Therefore  '  ^*  *^'  7.| 

Multiply  as  in  56,  a?id  2?r<^a;  ^/?e  sj^yi  -}-  io  the  product. 


EXAIMPLES. 

(1.)         (2.)  (3.)  (4.)  (5.) 

Ill 

Multiply         — X         — X         — 3a^6  — 2a^x^y^         — ^xyz 

12      3 

by  — y         ■ — X         — ^a^hx       — \a?-x^y'^         — Zxyz 


Ans,             xy             x^ 

(Vide  Def.  9,  last  clause.) 

(6.) 

(7.) 

(8.) 

(9.) 

Multiply             —  ^xy^ 

—  bxyz 

—  Ixy 

— ba^hc 

by                   —  ^ay^ 

—  ^xyz^ 

~2ah 

—  9axy 

59.  Hence,  to  multiply  algebraic  monomials : 

Jlidtijjly  the  coefficients,  and  add  the  exponents  of  like  letters. 
Like  signs  produce  -f- ,  a7id  unlike  signs  produce  — . 


EXAMPLES. 

(]-) 

(2.) 

(3.) 

Multiply 

—  hax  y  z 

5  a*  h^  c' 

. 5.t"    ?/^    z^ 

by 

4taxy  z 
—  20a''xhfz'' 

—  ?>n'  h^c^' 

■ —  3.T    y     z 

Ans, 

—  Iba^'O'^^c'' 

i^x'+y+^z'^^ 

MULTIPLICATION. 


So 


4.  Multiply  — lOa^x'y  by  Aa^x^/K  Ans.  —  AOax^f^ , 

5.  Multiply  7x^fz^  by  3x^i/^z\ 

6.  Multiply  llx^fz^  by  — 2x^f^z\ 

7.  Multiply  —  ISninj)  by  Grnnp. 

r.  111  345 

8.  Multiply  — 2Gx'^}/^z^  by  — ^^^x'^i/^zK 

9.  Multiply  12^3^^^^  by  12xi/z. 

10.  Multiply  13iCj/^  by  — 13xi/z. 

11.  Find  the  product  of  13x^y  X  =^^6^^^^  X  14:X}/z  X  f^jj/^. 

J.  715.    X^  }f  Z, 

12.  Find  the  product  of  7xyz,  — Sx^e^  and  l^'x^y^^. 

Ans.  —  2x^ifz^. 

13.  Find  the  product  of  la'h^c',  ^a}?c^  and  —\a^}?c, 

Ans.  —7a^+*h'-'+*c'+\ 

14.  Find  the  product  of  Sa'^h^z"  and  4a"* Z)''^^. 

^;j5.  12a^+'"Zy»'+^-^'^+-''. 

..    60.  To  multiply  a  polynomial  by  a  monomial: 

Multiply  each  term  of  the  multijilicand  by  the  multijylier,  accord- 
ing to  59. 


EXAMPLES. 


Multiply 
by 

Ans. 

Multiply 
by 

A7IS. 


(1.) 

3 

Sx-\Sy-\-3z 

(4.) 

x^-{-x^-\-  x^ 
i 

~i        3        1 
x^  -f  x^^  4-  x^ 


(2.) 
ic — y  —  z 
4 


(3.) 
X — y  —  z 
—  5 


4x — iy — 4cz         — bx-{-by-\-bz 

(5.) 

x^  —  2x'  4-  3a;y  —  77n  +  12 


ic'^ — 2x*y-{Sx^y^ — 7?7ixy-{-12xy 

6.  Multiply  x^-\-xy-\-y^  by  cc^  ^4/^5.  re'' -f- a;'y -}~  a:'^^ 

7.  Multiply  x^ -f"  0^3/ -|- y2  by  — xy.  A71S.  — x^y  —  x^y"^  —  xy^. 

8.  Multiply  x^  -\-  xy  -\-  y^  by  y^.  Ans.  x^y^  -4-  xy^  -f~  ^^ 

9.  Multiply  x^  -\-  x^y  -f-  xy^  -]-  y^  by  x. 

A71S.  X*  -["  ^'y  +  ^V  ~\~  ^y^' 


36 


U  U  L  T  I  P  L  I  C  A  T  I  0  N  . 


10.  Multiply  x^ -\- x~^ -{- a'?j-^ -^  7/^  by  — y. 

Ans.   —  x^y  —  x^ij^  —  xr/^  —  y*. 

11.  Multi^jly  x^  -\~  X*  -\-  x^  -\-  x^  -}-  X  -\-  1  by  x. 

Alls,  x^  +  x^  -\-  x^  -^  x^  -\~  X. 

12.  Multiply  x'  +  x'  +  a;'  +  x^  +  x  +  1   by  —  1. 

A^js.  —  x^  —  x'^  —  x^  —  x^  —  x  —  1. 

13.  Multiply  X  -{■  y  by  x.  Ans.  x^  -j-  xy. 

14.  Multiply  X -}- y   by  y.  A?is.  xy-[-y^. 

"    61.  To  multiply  one  polynomial  by  another : 

Multiply   every    term   of   the    midtij^lkand   hy  each   term  of  the 
multij)lier,  and  add  together  the  several  products. 


1,  Multiply 
by 


Product 

x^'  +  2xy-{-7f 

2.  Multiply 

^^  +  <'^I/+f 

by» 

x'^  —  xy-\-y^ 

x'-\-x'y-\-xY~ 

—  x^y  —  x^y~  —  xy^ 

^'y^-^^u'^y' 

Product 

X*            +a;y-            -l-y 

3.  Multiply 

x^  J^  x^y -\- xy^ -\- y' 

by 

X  —  y 

EXAMPLES. 

x-^y 

x'^-\-xy  (Vide  above,  ex.  13.) 

^y-^-y^    {Vide  above,  ex.  14.) 
{Vide  34.) 


(Vide  above,  ex.  6.) 
(Vide  above,  ex.  7.) 
(Vide  above,  ex.  8.) 

(Vide  34.) 


Product 


X* -\- x^y -{- x^y^ -\~  xy^  (Vide  above,  ex.   9.) 

—  x^y  —  xhf  —  xy^  —y^(  Vide  above,  ex.  10.) 

4 


X 


-r 


MULTIPLICATION.  37 

4.  Multiply  x'-^x'^-]-x^-\-x^-{-x-{- 1 

by        X  —  1 

x^  -\-  x'  -\- x^  -\-  x^  -{- x"^  -\- X         (  Vide  above,  ex.  11.) 
—  x^  —  x'^  —  x^  —  x~  —  X —  1  (  Vide  above,  ex.  12.) 

Product  x°  — 1 

5.  Multiply 

by 


Product 

6.  Multiply 
by 


a;2  -|-  5x  +  7 
xi  —  8x  —  3 

a;4_}-5a;3_|-7x2 

—  Sx'—4.0x'- 
—  3x"  — 

~  hQx 

-  15x-21 

x^  —  Sx'  —  ^Gx'- 

x^-{-xy-{-y^ 
x^  —  xy-\-  y^ 

-71a;~21 

—  x^y           — 
1   1 

^x^y^ 

■  x^y^  —  xy^ 

J^xy^-\-y 

Pi'oduct  X  -{-2x^y^—x'^y^         -\- y 

7.  Multiply  X  +  5  by  a;  +  G.  An%.  x^  +  11^^  +  30. 

8.  Multiply  cc  +  5  by  a;  —  6.  Am.  x^  —  x  —  30. 

9.  Multiply  X  —  8  by  a;  —  9.  An^,  x"  —  llx  +  72. 

10.  Multiply  2^2  4-  3.x  —  1  by  a;  —  5. 

Ans.  2x'  —  7^2  —  16x  +  5. 

11.  Find  the  product  of  (x  —  l)   (re  — 2)   (.-r  —  3)   (x  —  4.). 

Ans.  X*  —  10a;'  +  35.7)2  _  50^;  +  24. 

12.  Find  the  product  of  (a;^  — 2x+  5)   (x  +  !)•  -^"S- 

13.  Find  the  product  of  (x^  +  2xy  -f  y^)   (x^  +  2xy  +  if). 

14.  Multiply  a)2  -j-  2ax  +  7  by  a;'  — ax  +  5. 

15.  Multiply  x'  +  3.xV  +  3a'/  +  y'  by  x^  +  2xy  +  /. 
16    INIultiply  x2  4-  2.ry  +  y^  by  x^  —  2a-j/  +  if. 


ir. 

Multiply 

by 

Aiis, 

18 

Multiply 

by 

Alls. 

38  MULTIPLICATION. 

x^  —  3x^1/  -j-  3x7/^  —  y* 
x^  —  3xy-{-3xY  —  y^ 

x^  4-  ^^^y  +  Gx^y^  4-  4.XJ/*  +  y^ 

x^  —  4x^y  -j-  6x^^^  —  4:X?/^  +  y* 
X*  —  4:xy  4-  Qx^  —  4:X^y^  -f  y* 

19.  Multiply  x^  —  X*  -\-  x^  —  x^-{-  x  —  1  by  cc  +  1.     Ans,  x^  —  1. 

20.  Multiply  l—x-{-x^~x^  by  1 -{.  x -\- x* -{- xK 

21.  Find  the  product  of  (x  —  5)  (x  —  6)  (x  —  7)   (a;  +  8). 

Ans.  x*  —  10x»  —  37a;2  -|-  U6x  —  1680. 

22.  Multiply  x^-}-x^  by  aj^—a^^.  Ans.  x—'xK 

111  111 

23.  Multiply  x^--x^-\-  x^  by  x^  -f  cc^  —  x\ 

2  _T  1 

I  ^  ^?is.  X  —  x^-{-  2x^^  —  xK 

24.  Multiply  x^-{-^x^  —  x  by  x^  —  ix^-\-x. 

25.  Multiply  a:^  -j-  ^x^  -\~  Ix^  by  a;^  —  5a;^  —  3a;*. 

26.  Multiply  Ix^-^ix^  by  5aj^-f  6a;^. 

27.  Multiply  a:"  +  y**  by  a;"  +  ^^  ^?is.  a:^'' -f  2a:"^'*  +  ^2". 

28.  Find  the  product  of  x"^  —  3/"*,  a:*"  +  y"*  ^"^l  a;"  —  y\ 

Ans.  cc^"'+"  —  a:"^^'"  —  x'^"'y''  -\-  y^"^\ 

29.  Find  the  product  of  a;"  +  2a;'"y»  +  3^3/^  by  x"' —  y\ 

30.  Find    the    value    of   (x^  -j-  3/')    (a;  +  y}    (x  —  ?/)   in   a  single 

polynomial. 

31.  Find  the  value  of  (x*-^y^)   (^^4-^)   (x  +  y)   Q^—y)- 

32.  Find    the    value    of   (x^  —  xy-\-  y^)    (x^  -f  xy  -j-  y^)    (x  +  y) 

(p^—y)' 

33.  Find  the  value  of  (^c^  4.  5^;  -f  7)   (a;^  _  5x  +  7). 

34.  Find  the  value  of  (a:^  -f  6a;  +  18)   (a;^  —  Gx -\-  18). 

35.  Find  the  value  of  (a;^  _[_  4x  +  8)  (x^  —  4x  +  8). 

36.  Find  the  value  of  (a;^  +  2aa;  +  2a^)   (x^  —-  2ax  +  2«2-)^ 

37.  Find  the  value  of  (ar  +  3)   (x  +  4)   (x  —  5)   (;«  —  6). 

38.  Find  the  value  of  (a;  — 5)  (x  -  6)   (c--4). 


MULTIPLICATION.  39 

39.  Multiply  x-{-  ^  by  x  -\-  ^,  x  —  5  by  a:  —  5,  x-\-7  byic-f4, 
and  X  -{-  7  by  x  —  4. 

62.  Since  (x  +  y)  (x -\- y)  =^  (x -^  y)^  =  a;'  -f  2xy  -\-  y\  it  fol- 
lows that 

The  sqtiare  of  the  sum  of  two  quantities  is  equal  to  the  square 
of  the  first  +  twice  their  product  -f  the  square  of  the  last, 

EXAMPLES. 

1.  (a  -\-iy  =^  a"  -{-^ah  -\-h\ 

2.  (2a  +  If  =  4a2  +  4ah  +  h\ 

3.  (a  +  26)2  =  a2  +  4a&  4- 46^ 

4.  (3a  4-  2hy  =  9a2  +  12a6  +  4b\ 

5.  (x^  +  y^y  =  x-\-  2x^^y^  +  y, 

6.  (x^  -^-y^y^x^  -{-2x^y^  -\-y^, 

7.  (2x»  +  3x0'  =  43;'°  +  12x»  +  9x». 

8.  (x^  4- 5)2  =  x*  +  10x2  +  25. 

9.  Find  the  value  of  (x^  4-^)2,  (x2  4-^2)2,  (x'4-x)2,  (x' 4- x)«, 

(a;3  _|_  y\y^    (x  4-  4)^    (x2  4-  x')2    and    (x»  4-  x^',    and    the 
numerical  value  when  x  =  8,  y  =  8. 

63.  Since   (x—y)    (x  — ^)  =  (x  — y)'=    x'  — 2xy4-/,  it 
follows  that 

The  square  of  the  difference  of  two  quantities  is  equal  to  the 
square  of  the  first  —  twice  their  product  -\-  the  square  of  the 
last. 

EXAMPLES. 

1.  (a  —  hy^a'^  —  2a'b-\-lK 

2.  (2a  —  t)' =  4a' —  4a6  4- &2. 

3.  (a  —  25)2  =  a2  —  4a6  4- 4Z>2 

4.  (x  — 2)2  =  x2  —  4x  4- 4. 

5.  (1  —  4x2)2  =  1  __  8x2  _|_  i6x^ 

6.  (5  -  4)2  =  25  -  40  4-  16  =  1. 


40  ^I  U  L  T  I  P  L  I  C  A  T  ION. 

7.  (1  —  1)2  =  1  —  2  +  1  =  0. 

8.  (ox^  —  xy  =  25x  —  10a;2  -j-  xK 

9.  Find    the    numerical   value   of   (x  —  3)^,    (x-  —  5)',    (1  —  xy, 

(3^2  —  2^3)2,   (4x3  _  2xy,    (px^  —  2y)2  and   (x  —  4)-  when 

X=4:,     7J=1. 

64.  Since  (^  +  y)    (^ — y)  =  ^^  —  3/^  i^  follows  that 

T/ie  product  of  the  sum  and  difference  of  two  quantities  is  equal 
to  the  difference  of  their  squares. 

EXAMPLES. 

1.  (rt-f-Z>)   (a  —  h)  =  a^  —  l\ 

2.  (2a  +  h)   (2a  —  h)  =  4a2  —  h^ 

3.  (03  +  4)   (x  — 4)  =  a;2  — 16. 

4.  (x  +  5)   (x  — 5)  =  x2  — 25. 

5.  (5  +  2)  (5  —  2)  =  (25  —  4)  =  7  X  3  =  21. 

6.  (4  J-  _j_  4?^t )  (4x^  —  4?/^  )  =  1 6x3  _  I Q^i^ 

7.  (3x^  +  22/2- )   (3^i  _  2/  )  =  9x  —  4y.        / 

8.  (0)^'  +  /)    (x^-/)  =  x-7/. 

9.  Find    the    numerical    value    of   (x^-\-y^)    (x^ — y'),    (^'+y) 

(x' — 3/')  and  (x*-\-y^)   (x^ — ^0  when  x  =  5,  y  =  5. 

65.  Since   (x  +  ?/  +  21)2  =  x^  +  /  +  2^  +  2xy  +  2x.^  +  23^2;,  it 
follows  that 

The  square  of  a  trinomial  is  equal  to  the  sum  of  the  squares  of 
the  three  terms  united  to  twice  the  j^roduct  of  the  terms  taken 
two  and  tivo.      (For  the  signs  observe  59.) 

EXAMPLES. 

1.  (a  +  &  +  c)2  =  a2  ^  2,2  _|_  c2  _|,  2al  +  2ac  +  2hc. 

2.  (a;  +  3/  —  2)2  =  x2  +  y2  _|_  ;j2  _^  2x3/  —  2o;;2  —  23/^. 

3.     (-3,_y_^)2^^2   _|_^2    _|_  ^2  _  2.xy  _  2X^  +  23/-^. 

4.    (a;2  _|_  a;  _}_  1)2  =:  a;4  ^  a;2  _^  1  _|_  2x3  _|_  2x2  _^  2x  =  x^  +  2x' 
+  3x2  +  2x  +  1. 


IM  U  L  T  1  P  L  I  C  A  T  ION. 


41 


5.  (5x2  _  4a;  _j_  3)2  =  25x^  —  40^^  +  46^^  —  24.c  +  0. 

6.  (3x'  4-  4x2  _j_  5.,.-)  2  ^  9,^6  _|_  24.^:5  _i_  4Cx^  +  40x^  +  2ox\ 

7.  Find  the  value  of  (Ox^  —  6x  — 2)^,   (x^ -f  y^  _|_  ^y^2^  ^^_|_2^ 

-\-^z)^  and  the  numerical  value  when  x  =  4,  ^=9,  2=  16. 

8.  Find  the  values  of  (^x -\- 4:i/)^,  (3x  — 4j/)2,  (3x  +  4y)   (3x — 

4y),   and    (1  +  3x  —  4?/)^,  and   the   numerical  values  when 


X 


9 


2,^=1. 


66.   1.  Multiply  (x  —  a),   (x  —  6)   and   (x  —  c)   together. 

(^Vkle  Def.  12.) 


X  —  a 

X  —  h 

X  +  ah 

X? — a 

—  h 

X  —  c 

x^ — a 

x^  +  «c 

X  - 

-  (the 

^h 

4-  he 

—  c 

-\-ah 

Ans. 


2.  AYhat    would    be    the    answer    to    the    last    example  when 
a  =  1,  h=2,  c  =  3  ?  Ajis.  x'  —  Gx'  +  llx  —  6. 

3.  AVhat  would  be  the  value  Avhen  a  =2,  h  =  3,  c  =  4? 

Ans.  x'  —  9x=  +  26x  —  24. 

4.  Find  ihe  product  of  (x  —  a)  (x  —  h)  (x  —  c)  (x  —  d),  and 
what  will  it  become  when  «=1,  h  ==  2,  c  =  3,  f/ =  4 ;  also 
when  a  =  3,  ^>  =  4,  c  =  5,  (/=6.  (Vide  61,  ex.  11.) 

^l«s.  to  last  x'  —  ISx'  +  119x=  —  342x  +  360. 


42  DIVISION. 


DIVISION. 

6t.  Division  in  Algebra  is  the  operation  o?  finding  a  quotient 
which  multiplied  into  a  given  divisor  will  produce  a  given  dividend. 

6S.  To  divide  one  monomial  by  another : 

By  56   we  have   7a^6  X  5ax  =  35a' tx  .*.  ^ha^hx  -~7a?b 
=  bax. 

By  5t  we  have  4a'Z>  X  — 2a'Z;r=  —^a'h^  .-.  —8a*?*' -^4 

i  1  2  1 

Also,  (example  10)  3.x'^y~' X  Sx'^y  =  ISx'^"'  .*.  l^x'^ y~^ 
1  1 

-T-  5x'^y  =  3x^?/~% 

Hence, 

(1.)  Divide  the  coefficient  of  the  dividend  hy  that  of  the  divisor ^ 
and  annex  all  the  letters,  giving  to  each  an  eoqjonent  found 
hy  subtracting  the  expo7ient  of  a  letter  in  the  divisor  from  the 
exponent  of  the  same  letter  in  the  dividend. 

(2.)  If  the  dividend  and  divisor  have  like  signs,  the  quotient 
is  iilus. 

(3.)  If  the  dividend  and  divisor  have  unlike  signs,  the  quo- 
tient is  minus. 

(4.)  If  the  divisor  contains  letters  not  found  in  the  divi- 
dend, these  letters  may  be  inserted  in  the  dividend  by  giv- 
ing to  each  the  exponent  0.  (Vide  Def.  13,  9.) 
Thus,  4x2  =  4a»?>»c"x^ 


ahc           a h c 

EXAMPLES. 

1. 

a           0 
a 

1. 

2. 

2                     a' 

a^  ■—  a  ■=■  -  =  a. 

a 

3 

3     .                  '^               2 

(Vide  Def.  13,  9.) 


DIVISION.  43 

4  9         a  1  1 

«2         a 

«  0         (, 

5axy         y  .      i 
.  aa; -T-axy  = -^=- =  y     . 

•^         axy        y        ^ 

6.  10a2x'-^5a-•ic^  =  2a^x-^ 

7.  —  lOOxV^  -T-  —  SOrcy  ^2  =  2a;->^-'2-'. 

8.  7xi/-^6?n=:'^-^  =  lxi/m-K 

9.  21a3Z;5_|_7a2Z>4^  57a^62_^19a5,  l^aHx-^3a,  S^a^h^x'' -^7a. 

10.  27a^Pc'-^9ac,   —33x'f-i-llx7/,    —  42a:y -r- 21xy,   4:bx* 

y^  H-  ISary. 

11.  7x^y^-T-  —  3^xy,  —20x^y-^^xy\  3a?-~2h,  7ahnn-^3aK 

12.  —  26a;* 3^  -. l^xy,  16xy  h-  8xy,  —  30xy^  -i-  IS.-cy,  72x^y^  -~ 

36x'*y^. 

Ill        111        111 

13.  x^y^z^ -T-x^y^z^  ==  x^y^z*^, 

I    t  1  111  i.i_i 

14.    lOiC^J^^  Z^  -T-  5x6^3  23  _  2x30^3  g;-  1  2 

15.  --6lxy'-i--2lxy*  =  3y. 

16.  —  7|rc*3/  -7-  —  2-J^xy  =  Ba;^ 

17.  729a;^ -T- —  27ic^^^,    —22^x^y^-i-  —  lbx~^y^,    —12ixhji 

18.  441a;2y  -^  —  21ic/,  361x'y^  -r-  —  19a;«/,  ISary  -i-  l^xyz, 

69.  To  divide  a  polynomial  by  a  monomial: 

Divide  each  term  of  the  dividend  hy  the  divisor,  according  to  68. 

EXAMPLES. 

1.  Divide  x^  +  ^^y  +  v'^  ^J  ^-  -^*^s.  cc  -f-  2y  4-  3/^-'^~'» 

2.  Divide  3aa;^  +  Sa&ic  —  9x'  by  3a;.  -4?is.  ax  +  2a6  —  3a;^. 

3.  Divide  xy  -\-  xz  -\-yz  by  xyz.  Ans.  z~^  -{■  y~^  -\-  x~\ 

4.  Divide  xyz  -\-  xyw  +  0:21^  -j-  yzw  by  ccy^zy. 

5.  Divide  —  6x*  +  SOoj'  —  9aa;'  -f-  Qirix  —  3nx  by  —  3.T. 

6.  Divide  12a^  +  24a!'  —  ^Qa"  by  12a.     Ans,  a^-^  +  2ay-^  —  3a'-\ 

7.  Divide  4:x\x -\- yy -~Sx\x-\- yy -{- 12x\x+ y)'  by  4:xXx+yy. 

Ans,   1  —  2x  (x+y')  -f  Sx^  (re  +  yy. 


44  D I V 1  tj  I  o  X . 

8.  Divide  7x^  (x  +  3^  -j-  .)  +  4.r ^  (.r  +  y  +  ~)  by  3x^  (x  +y+  2;). 

Ans.  2^  +  f.'ci 
•iO.   To  divide  one  polynomial  by  another: 

(1.)  Arrange  the  polynomials  uith  reference  to  the  ascendvig  or 
descending  poicers  of  the  same  Utter, 

(2.)  Divide  the  first  term  of  the  dividend  hy  the  first  term  of 
the  divisor  for  the  first  teim  of  the  quotie?it.  Multiply  the 
whole  divisor  hy  this  term  of  the  quotient,  and  subtract  the 
2)roduct  from  the  dividend  for  the  first  remainder. 

(3.)  Z^pon  the  first  remainder  repeat  the  very  same  operation  as 
vpon  the  given  dividend,  for  the  second  re7nainder,  and  so  on 
till  there  is  no  remainder ;  or,  if  the  division  he  not  exact,  till 
the  first  term  of  the  remainder  divided  hy  the  first  term  of 
the  divisor  ivould  produce  a  negative  exp)on€nt  in  the  quotient. 
In  this  case  p)lace  the  remainder  over  the  divisor  at  the  Hght 
of  the  quotient,  prefixing  the  j^^^ojJer  sign. 

(4.)  It  often  happens  that  the  division  may  be  carried  on 
indefinitely,  giving  rise  to  an  infinite  seizes,  in  "which  case 
a  few  of  the  leading  terms  of  the  quotient  will  generally 
be  sufficient  to  determine  the  rest. 

EXAMPLES. 

1.  Divide  x^  -f  --^j/  +  !/^  by  x  +  y- 

Solution. 
Dividend      x^  +  2xy  -j- y''  \  x -j- y  =     Divisor.        (Vide  Def.  11.) 
x^  +    xy  X  -}-  y  =     Quotient, 

^'^  +  f 


Kiplanation. 

The  first    term,  x^,  of  the   dividend    is   divided   by  x,  the   first 
term  of  the  divisor,  and  we  obtain  x,  the  fir^t   tcnn  of  the  quo- 


DIVISION.  45 

tient.  Next,  multiply  the  divisor  x  ~{- y  by  x,  and  we  obtain 
the  product  x^  -\-  ccy,  which,  subtracted  from  the  dividend,  gives 
xy  +  y"^  for  the  remainder.  Divide  its  first  term  xy  by  £C,  and 
we  obtain  y  for  the  second  term  of  the  quotient.  Multiply  x-\-y 
by  y,  and  we  have  xy  -f-  y^,  which  being  taken  from  the  remain- 
der leaves  nothing,  the  quotient  being  exact.     (Vide  62  &  61.) 

2.  Divide  cc^  —  2xy  -{•  y^  by  x  —  y.  Ans.  x  —  y. 

3.  Divide  x^  +  '^x^y  +  Zxy^  +  y^  by  x^  -\-  2xy  -f  y^.    Ans.  x  -{-  y. 

4.  Divide  x^  —  3x^y  ~\-  Sxy^  — y^  by  x  — y.     Ans.  x^  —  2xy  -f  y^. 

5.  Divide  x*  +  'ix^y  +  6x^y^  +  4x/  +  ^  by  x^  +  2xj/  +  y\ 

Ans.   x^  -\-  2xy  -j-  y^. 

6.  Divide  x"^  —  S.-c'  —  SGx^  —  Tla?  —  21  by  x^  —  8x  —  3. 

Dividend,    tc^  —  3x3  _  3(3^2  _  ^i^  _  2I  |  x^  —  8x  —  3      Divisor. 

ic^  +  5x  +  7    Quotient. 


tc^  —  8x3 . 

-3x2 

5x3 

—  33x2  — 71x  — 21 

5x3 

-40x2-15x 

7x2  _  5(3^  _  21 

7a;2  _  56x  -  21 

7.  Divide  x3  —  x^  -j-  3x  4-5  by  x  +  1.  Ans.  x^  —  2x  -f  5. 

8.  Divide  4x«  —  25x2  ^  oQx  —  4  by  2x3  —  5x  +  2. 

Ans.   2x3  _|.  5,^  _  2. 

9.  Divide  2x3  _  7^2  _  ig^  _|_  5   by  x  —  5.      Ans.  2x^  +  3x  —  1. 

10.  Divide  x^  —  3x^y^  -j-  Sx'y*  —  y^  by  x'  —  3x2y  -j-  3xy^  —  y^. 


46  DIVISION. 


Operation. 

x^  —  ?>x'^y'^  -\-  Sx^y^  —  y^    I  x^  —  3x^y  -j-  3xy^  —  y^ 
x^  —  3ic*y  +  ^x^y^  —  x^y^     x?  +  3x^j/  -f  ^ixy"^  -\-  y^ 
Zx^y  —  Gx"*  2/^  4"  ^y^  +  Sx^^**  —  y^ 


Zx'^y^  — 

-8x-^y  _|_  6.xy  —  3/« 

Zx^y""  - 

-  9x'^3  _^  9^2y  _  3^^5 

x^y^  —  3x^3^^  4-  3x^^  — y* 
x^y^  —  3x^y  +  3xy'  —  y^ 

11.  Divide  x^  —  2x^y^  -\-  y""  by  x^  —  2xy  +  y"^, 

Anu.  X?  -f  2xy  4-  3/^. 

12.  Divide  x^  —  ^x^y^  +  6x^ y*  —  4x^^^  +  ^^  by  x"  +  4x'y  +  6x* 

2/^  +  4x2/'  +  3/''.  ^7is.  x"*  —  4x^y  -f-  6x'^^  —  4x^'  +  y^, 

13.  Divide  x^  +  x^^^  +  ^"^  by  x*  +  xy  +  ?/^      J.?is.  x*  —  xy  +  y^, 

i  ^         ,  ,  ^  1 

14.  Divide  x  +  2x^3/^  _  ^2^2  _j_  y  j^y  ^^2  __  ^^  _j.  ^2^ 

Operation, 

11  11 

—  x*3/'  +  ^  +  2x2^2  _j_  y      —  icy  _|_  aj2  _|.  y2 

33  11- 

—  x^y^  +  '^^y  +  ^3/^  ^y  -\-  ^^  ~\-  y^ 

i  3  1  j 

—  x^y  —  xy^  -^  X  -}-  2x2^2  _|_  y 

3  11 

—  x^y  +  a^  +  -'^^3/^ 


3 

1    1 

-X3^2 

+  X2^2  _^  y 

3 

1      1 

~X^2 

+  x^y^-i-y 

211  11 

15.  Divide  x  —  x^  by  x^  +  x^.  ^?js.  x^  —  x', 

16.  Divide  x  —  x^  +  2x^2  —  .x^  by  x^  +  x^  —  xl 

i         1         1 

J.?2S.    X^  —  X^  -f-  X^. 

17.  Divide  x  —  l^^ajg^  ^x*  — fx^  —  |xi2  — sx^  by  x^  — 5x5  — 3x^. 

.Ill 
Ans.  x^  -\-  |x3  -}-  ix'^ 


DIVISION.  47 


1        J       1 

A  i.  x^-\-if  —  z\ 


18. 

Divide 

X 

2                 111                  1 

_  ^3  _}_  2^3  2;^  —  z-  by  x^  — 

1 

^3 

1 

19. 

Divide 

x' 

+  5x  +  7  by  X  —  3. 

Oj~teration, 
a;2  +  5x  +  7      .c  —  3 

ic2  _  3a;              a;  +  8 

+ 

31 
z-3 

8x  +  7 

8a;  -  24 

31 

14 

20.  Divide  x'  —  6a;  —  41  by  a;  +  3.  Ans.  a;  —  9  —  — _^. 

21.  Divide  a;'  +  4a;2  +  5a;  +  6  by  a;'  +  3a;  —  1. 

3x  +  7 
^«S.    X  +  1  +^2  +  3^_i 

22.  Divide  1  +  3/'  by  1  +  a;. 

l_px       l__aj_|_ic'  —  aj'+  &c. 


—  a; +  3/2 

—  a;  —  a;' 

x^-^y^ 

x^  +  a;' 

-a;' 
-a;' 

23.  Divide  1  +  2a;  +  Sx'  by  1  —  4x. 

Ans.  1  +  6x  +  27x2  +  lOSx'  +  432x^  +  &c. 

24.  Divide  1  —  x  by  1  +  x.  Ans,  1  —  2x  +  2x2  _  o^z  ^^^ 

25.  Divide  1  —  x  by  1  +  x  +  x^. 

Ans.  1  -  2x  4-  a;2  +  a;'  -  2x''  +  x*  +  x«  -  2x^  &c.. 

26.  Divide  1  +  2x  by  1  —  x  —  x^. 

=«-«^  •  A71S.    1  +  3X  +  4x2  _|,  7^3  _^    ^c. 


48  DIVISION. 

27.  Divide  x^"  -f  x'-'f"  -f-  i/"  by  x^"  +  .T"y  -f  ?,^'\ 

Ojjej'alion. 

^4rt  _j_  ^^2,1  ^2»  _|_  ^An        I   _^2«  _^  ^n  ^n  j^  yin 
y  _J_  ic^^y^  -f-  a;2"  ?/2,i    rf.%1  _  rf-nyn  _j_  ^2,1 


X 


_^  ^3}i  o.ji  ___  /y,2/i  o/2;i  ^__  ^,»i  j/3/i 


28.   Divide  x""^  +  2x2"*  y'  _|_  ^m+i^p  __  ^^^^n  __  2.x"' ^^n  _  ^^^-^n  ^^^ 

^m  _  ^n^  ^.,^5^    ^n  _^  2x'"^''  +  X^/^ 

"  71.  1.  Divide  x'  — y^  by  x  — y.  Ans.  x^  -\-  xy  -f  ?/*, 

2.  Divide  x^  —  y"^  bj  x  —  y.  ^7i5.  x-\-y. 

3.  Divide  x'  -f-  ?/^  by  x -\-  y.  Ans.  x^  —  xy  +  y^ 

4.  Divide  x^  —  y^  by  x  +  y.  A?is.  x^y 

5.  Divide  x^ -\- y"^  by  x  —  y.  Ans.  x -\- y -\    ^^ 


6.  Divide  rc^  -{■  y^  '^J  x—y.  Ans.  x^  -\-  xy  -\-  y"^  -{- 


x  —  y 
2y3 


x  —  y 


22/ 


7.  Di\ide  x^  —  ^'^  by  x  +  y.  Ans.  x^  —  xy  -\-  y^  ^ 

"^  x  +  y 

8.  Divide  x^  -\-  y^  by  x  +  y.  Ans.  x  —  y  ~\.  -^ 


x  +  y 

(1.)  For  the  present  we  may  infer  from  1  and  2  that  the 
difference  of  two  quantities  ivill  divide  the  difference  of  the  like  powers 
of  those  quantities,  tvithout  a  remainder.     (^Vide  I'S.) 

(2.)  From  3  and  4,  the  sum  of  two  quantities  will  divide  the 
sum  of  the  like  odd  or  difference  of  the  like  even  powers  of  those 
quantities,  loithout  a  remainder.     {Vide  ^9,") 

'  (3.)  From  5  and  6,  the  difference  of  two  quantities  will  not 
divide  the  sum  of  the  like  powers  of  those  quantities,  witliout  a 
remainder. 


DIVISION.  49 

(4.)  From  7  and  8  the  sum  of  two  quantities  will  not  divide 
the  differe7ice  of  the  like  odd  or  the  sum  of  the  like  even  powers 
of  those  quantities,  without  a  remainder. 

'  "ys.  All  the  above  cases  may  be  solved  mentally  by  observing 
the  following  directions: 

(1.)  The  exponent  of  the  leading  letter  aj  decreases  by  one  reg- 
ularly, and  this  letter  disappears  in  the  last  term. 

(2.)  The  exponent  of  y  increases  by  one  regularly  from  the 
second   term. 

(3.)  If  the  divisor  contain  the  negative  sign,  then  the  terms  of 
the  quotient  will  all  be  positive. 

(4.)  If  both  terms  of  the  divisor  are  positive,  then  the  odd  terms 
of  the  quotient  are  positive,  tlie  even  terms  negative, 

(5.)  In  the  cases  which  have  a  remainder,  this  remainder  will 
always  be  twice  the  given  power  of  y,  retaining  its  sign. 

EXAMPLES. 

1.  Divide  x^  —  y^  by  x  —  y.  Ans,  x*  -\-  x?y  -{•  x? y"^  -\-  xy^  -f  3/^« 

2.  Divide  x*  -- y^  by  x  —  y.  Ans.  x^  -\-  x'^y  -\-  xy"^  -j-  ?/'. 

3.  Divide  x^  -{■  y^  by  x  -\-  y.  Ans,  x^  —  x^y  +  ^^j/^  —  ^y^  +  y^. 

4.  Divide  re''  —  y^  by  x  -\-  y.  Ans.  x^  —  x^y  +  ^y"^  —  y^. 

5.  Divide  x'^  -\-  y^  by  x  —  y,  Ans.  cc'  +  x^y  +  xy^  +  j/^  +  ~~-  • 

X     y 

6.  Divide  x^  -]-  y^  hy  x  —  y. 

Ans.  x^  +  x^y  -\-  x-y'^  +  ^^^  +  3'''  +  — ^  • 

7.  Divide  x^  —  3/^  by  x  +  y. 

Ans.  x^  —  x^y  +  x^y^  —  xy^  -\-  y^  —     ^ 


8.  Divide  x^  -\-  y^  by  x  -f-  y.       Ans.  x^  —  x'^y  +  xy"^  —  ^^  + 

9.  Find  the  value  of 


x  +  y 
2yt 


x  +  y 

X  —  y     x^  —  y^     x'  —  y^     x^  —  y^     '^  -\-  y 


X 


y'    X  --  y'    X  —  y'     x—y'    x  -\-  y 


^1±J^  and  '^±^. 

^-\-  y         ^  +  y 


50  DIVISION. 

10.  Find  the  value  of -^,  -^, ^, ^,  —T-J-~ 

'^-\-  y     •^'  +  ^     •'^'  +  ^     ^  +  y     ^—y 

and . 

X  ■\-  y 

11.  Divide  x"' —  y"^  by  x—y,     Ans.  x'''~^ -\- x"'-'^  y -\- x'^'-^y^   &c. 

. .     ^.    ,      ,  .  ^    .-«'  —  1      .-c'  —  1      0^5  +  32     x'  —  2'i 

12.  Find    the    vahie    ot    — ,    — ,  ^  ,    ^, 

cc  —  1       X  —  1        X  -{-  2        X  —  3 

,      .7;*  4- 2-43        ,  81rc<  — icy      / 
t  ^   ,^      and  -^r — ^.    f 

.x  +  ^  ^^  —  ^y 

To  find  the  answers  of  examples  12,  make  in  11,  ?/i  =  2, 3/  =  1, 

m  =  3,  y  =1;  m  =  5,  ?/  =  2  ;  m  =  3,  3/  =  3  ;  ?7z  =  5,  ?/  =  3  ; 
»?i  =  4,  and  place  in  the  last  3.^;  for  x  and  2y  for  ?/,  using  as 
many  terms  of  11  as  is  indicated  by  the  value  of  m, 

Ans,  to  the  last  27rc^  +  ISx^y  +  12xy^  +  8/. 

13.  Divide  a;'""  — ^y*^"  by  x^  —  y\ 

14.  Find  the  value  of  —. ^,  ~ -.,—, -,  and  —^ V- 

^  —  ^     ^  —  ;^     ^  —  y  ^  —  ^ 

15.  If  in  ex.  1  cc  =  y,  what  does  the  answer  become  % 

Ans.  5x^  or  5^"*, 

16.  If  in  ex.  2  a?  =  y,  what  does  the  answer  become? 

Ans.  4x^  or  4^^ 

17.  If  in  ex.  11  cc  =  y,  what  does  the  answer  become*? 

Ans.  mx^"~^  or  wy^^^. 

18.  If  in  ex.  16  and  17  £c  =  5  and  m  =  4,  what  will  they  be- 

come? Ans.  500. 


>^hri'C.     fj  C>  ei  t  ^  ^' 


CHAPTER    III. 

FACTORING — GREATEST   COMMON   DIVISOR — LEAST    COMMON 

MULTIPLE. 

FACTORING. 

73.  Factoring    is    the    operation    of  resolving  a  quantity  into 
factors. 

1.  A    composite    quantity  is    one    whicli    may  be    resolved    into 
factors. 

2.  A  ^)?T?«e  factor  cannot  be  resolved  into  other  factors. 

3.  All  the  cases  of  factoring  merely  reverse  the  operations  of 
multiplication. 

74.  To  separate  a  monomial  into  its  prime  factors : 

Separate   the   coefficient    into   its  prime   factors,    and   annex    the 
literal  part,  also  resolved. 

EXAMPLES. 

1.  Find  the  factors  of  12a'^hx.  Ans.  2.  2.  3.  a.  a.  h.  x. 

2.  Find  the  factors  of  lQa\  l^x^y,  IVIx^y^  and  133a;2/. 

75.  To  factor  a  polynomial  when  multiplied  into  a  monomial : 
Divide  the  polynomial  by  tJie  monomial  common  to  all  the  tei^ns. 

The  divisor  and  quotient  are  the  factors.     (^Vide  60  and  12.) 

EXAMPLES. 

1.  Find   the  factors  of  a;^ -f  ^^«  ^'^^'   G'^  +  ^)  ^« 

2.  Find   the  factors  o^  xy -{- y-.  yins.   (^ -h  I/)  !/• 


52  r  A  C  T  O  11 1  N  G  . 

3.  Find   the  factors  of  ax  -j-  Ijx.  Ans.   Qi  -|-  l>)  x. 

4.  Factor  ax  +  ^^  +  c.t,  a^x^  +  Ir  x',  5x-j-  2t^.x  and  ox^  ij  -\-  Zxip, 

5.  Factor  x^y  —  x^}f-  +  x^f^  x^ y^  -\-  x^  if  and  5ax  —  26.T  -|-  ''^'« 

6.  Factor  .x  +  ax  —  2Jjx^  6hx  +  .x  —  ax   and   3orx  -j-  GZjx  —  12cx, 

7.  Find  the  factors  of  x™-'y  ^3/"*.  ^«s.   (x""^' — y""'Oy* 

8.  Find  the  factors  of  x  -\-  y  -{-  (I  -\-  m)  (x  -\-  y). 

Ans.  (1  -|-  Z  -|-  ju)  (x  -|-  y)' 

9.  (x  +  a)  (x  +  Z>)  =  x^  +  (a  +  Z>)  X  +  a6. 

10.  (x  —  a)  (x  —  6)  =  x^  —  (^a-{-h)x-\- al. 

11.  (x  —  a)  (x  -[-  &)  =  x^  —  (a  —  ?>)  X  —  a&. 

12.  (x  -j-  «)  (^  —  ^)  =  ''^^  H~  (^  —  ^)  ^  —  ^^• 

•ye.  To  factor  expressions  of  the  form  x"^ -}- 2xy -\- y^ : 

Take  the  square  root  of  the  extreme  terms,  and  place  the  sign 
between  the  roots,  that  is,  hefore  the  middle  term.  The  result 
is  one  of  the  equal  factors.     (^Vide  62  and  63.) 

EXAJIPLES. 

1.  Find  the  factors  of  x' -f  2xy  +  yl  Ans.  (x-\-y')  (x-\-y). 

2.  Find  the  factors  of  x'  -\-  2x^y''-^y\     Ans.  {x^-\-y^)  (0:=  +  ^=^). 

3.  Factor    64x'«  —  96x»  +  36x^    1  +  2x  +  x^,    1  _  2x  +  x^    and 

1   —  2X3  _^  a;3. 

4.  Factor  1  +  '-^'+^,  ^'-2  +  4>  1  +  ^'  +  1^^' ^nd  1 +  1  +  -^. 

5.  Factor  x  +  2x^y^  -\- y,  x^  —  2x^y'^  -]-  y\  25x  —  20x2  -J-  4x'  and 

2  4 

x3  —  2x  -]-  x\ 

6.  Factor  1  +  -  +  ^,    x^  — Sx+^f,    a;^ -f- 14x  +  49    and    x'  — 


x^ 


|X  +  ^ 


64  • 

'1'^.  To  factor  expressions  of  the  form  x^ — y": 

Indicate  the  product  of  tlie  sum  and  the  difference  if  the  square 
roots  of  the  quaiitilics.     {Vide  64.) 


L 


F  A  C  T  O  III  N  G  .  53 

EX  A:\irLES. 

1.  Factor   c'^  —  ?/'.  Ans.   (x -{- 1/)  (x  —  t/). 

2.  Factor  dx^~—4i/K  Ans.  (3x  -f  2^/)  (?jx  —  2>/) 

3.  Factor  1  — 4a;2,   l  —  9x%  4  —  1 63/2,   Ox'^  —  4i/^  and  1—-. 

4.  Factor  x'^  —  a;l  u4^?s.  a:2  (a:2  _  1)  =  a;2  (.x  -f  1)  (.t  —  1). 

5.  Factor  x'^  — y. 

Ans.  (x'  +  7/2)  (^2  _  j/2)  =  (.^2  _^  ^.^)  (-^  j^  y-^  {x-y). 

6.  Factor  x^— /,  x'^—y'%  x^'—f%  16x^  —  16^  and  IGa:"  — 81^. 

22  11  11 

7.  Factor  cc — y,  x^ — y^,  4^2 — y,  a:^ — y^  r^j-,^]  ^3  —  7^3^ 

8.  Factor  (x-^-jiy  —  cj^.  A7is,   (x -}- j) -\- q)  (x -\- ]) — q). 

•^S.   To  factor   expressions  of  the  form   x""  —  y'",  m  being  any- 
positive  whole  number : 

/y-m  o.TO  <v.TO 1   n,m — 1 

Since ^  =  x"'-^-^y- -^ ,   (vide   ^^'5,  ex.  7), 

X  —  y  X  —  y 

it  follows  that  when  x'^~^ — y'^"^  is  exactly  divisible  by  x — y 
then  x^ — y'^  must  also  be  divisible  by  x — y;  but  x" — y^  is 
exactly  divisible  by  x — y,  the  quotient  being  x -\- y  .*.  the  dif- 
ference of  tAvo  quantities,  &c.     (Vide  "71,   (1)  .)     Hence, 

To  factor  x"^ — y'",  divide   hy  x — y  and  indicate  the  product  of 
the  divisor  and  quotient. 

•^9.   To  factor   expressions  of  the  form  x"'' -\- y^,  m  being  any 
positive  odd  number : 

Smce  ^-^^  =  x"^^  —  x'^-^y  4-  ?/2 ^-^ , 

it  follows  that  when  x'""^  -[-  y'"~^  is  exactly  divisible  by  x  -\-  y, 
then  X'" -j-y"*  is  divisible  by  x-\-y\  but  x^-\-y'^  is  divisible  by 
aj-f-y?  the  quotient  being  x?  —  xy -\^  o?  .'.  the  sum  of  any  two 
quantities  will  always  divide  the  sum  of  the  like  odd  powers  of 
tlie  same  quantities.      (  Vide  '^1,   (2)  .) 

In   the   same   way  we   may  show  that   the  sum  of  two   quan- 


54  FACTOKIXG. 

titles  will  always  divide  the  difierence  of  the  like  cren  powers 
of  the  same  quantities.     Hence, 

To  factor  5c*-ry"  when  m  is  odd,  divide  hj  x-\-y  and  indi- 
cate the  product  of  the  divisor  and  quotient. 

To  factor  a:* — y*  when  m  is  even^  divide  br  x-\-y  or  by 
X — y,  &C. 

EXAMPIXS, 

1.  :^-f=(^x^y){:x^—xy^f), 

3.  x'-f=(2^^f)  Q^-y")  =  (x^y)  (x^-xy^f)  {x~y) 

(x^^J^xy^f). 

4.  x'  —y*  ={xr^f)  (x-\-y)  {x  —  y),  or   (x  —y)  (x^  -  x^y  4- 

^/-f/)»  or  (x-^y)(x^  —  x'y^xy^—y^). 

5.  Factor  sf—y^,  ^  +  y',  ^'—f,  x''—y''  and  x'^  —  y'"". 

6.  Is  7?-j-y^  a  composite  quantity?  , 

7.  In  how  many  ways  may  x? — y'  be  factored? 

An,.  7?-f=  (x^  -^y")  (a^  +  y)  (X  4-y)  (x  -y), 
x'-y'=  '>  —3^)  {x'  4-  xV  -f  a:^/,  &c. 

^'  —J/'  =  (-^  4-y)  (^'  —  ^'i/  -r  ^'i^2  _ ,  &c. 

x^-y^=f:c'J^y^)ix'-x^f^x'y^-y^. 
a:»-y»=  Or=-/)  {x' +  x' y'  -^  x^  y^  J^  y^)  =  (^x  ^  y) 
(x  —y){x^^  x^y^  4-  x^y^  -f  3^^}. 

8.  Find  all  the  factors  of  x^"^  —  t/'^  and  x^^  —  y^^. 

80.  To  factor  an  expression  of  the  form  x"^-^  (a-^lt)  x -\- ah, 

a  and  h  being  positive  or  negative  whole  numbers,  (lide  ^5,  ex.  { 

9.  10,  11  and  12)  :  J 
Find  by  inspeclion  a  and  h,  and   indicate   the  factors  thus :   (x  -f-  ,j 

a)(x-]-h),  ' 

EXAJEPLES. 

1.  a^  4-  9a;  4-  20  =  (x  -f  5)  (x  +  4). 

2.  x^  —  Ox  -f  20  =  (x  —  5)  (x  —  4). 


GREATEST      COMMON      DIVISOR.  00 

3.  x^-{-x  —  20=(x^5)  (a:  — 4). 

4.  a;2  —  a:  —  20  =  (.r  —  5)  (a;  -f  4). 

If   the    signs   of   the    trinomial   are    all    positive   the   factors    are 

positive. 
If  the  middle  sign  onli/  is  negative  botli  numbers  are  negative. 
If  the  last  sign  only  is  negative  the  largest  number  is  poskice. 
If  the  middle   and   last   signs  are   negative   the   largest  number  is 

negatice. 

5.  Factor  .r'  — 13.c -|- 42,  a;' -j- 13.r -^  42,  x^  —  x  —  ^2  and  a;'-f- 

X  —  42. 

6.  Factor   x^-\-x  —  l'l,  x''  —  x  —  V2,  x^  —  x  —  Zi)    and   x^~2x 

—  360. 

7.  Factor   ^i-^ -f  11.t -f  24,    x'  —  Zx^  —  -^,   x^-^10x^-\-^    and   x* 

—  3.T^-h2. 

8.  x'  —  10.r=  ~  9  =  (x'  —  9)  (x'  —  1 )  =  (x  -f-  3)  (x  —  3)  (.^  -f-  1 ) 

(x-1). 

9.  x'  —  17a;=  -f  16  =  (x  -f-  4)  (.t  —  4)  (a:  -f  1)  (a:  —  1). 

10.  Factor  x*  —  37x'  -f  36,  x^  —  26a:'  -{-  25  and  a:^  —  40a;'  -f  144. 

11.  Factor  4.r=  —  -ix  —  SO.  Ans,  4  (x  —  5)  (a;  +  4). 

12.  Factor  7a-'  — 7x  — 84,  ox^  -  5x  — 60  and  a;' —  13a:' -f- 42x. 


GREATEST    COMMON    DIVISOR. 

SI.  The  Greatest  Common  Divisor  of  two  or  more  quantities 
is  the  greatest  quantity  that  will  divide  each  of  them  without  a 
remainder.  Thus,  xy  is  the  greatest  common  divisor  of  a'y  and 
xf. 

8*2.  To  find  the  greatest  common  divisor  of  two  or  more 
quantities : 

Find  the  product  of  the  prime  factors  common  to  all  the  quantities. 


56  GREATEST     COMMON     DIYISOK. 

EXAMPLES. 

1.  Find  the  greatest  common  divisor  of  CL^h,  aJ?  and  ahc. 

Ans.  ah, 

2.  Find  the  greatest   common   divisor  of  3x^i/  -j-  3xi/^  and   C.x^?/^ 

-[-  6x^2/.  Ans.  3x7/, 

3.  Find  the  greatest  common  divisor  of  x^ -^  x  —  6  and  x^-{-Sx 

-{-15.  Ans.  x  -{-  3. 

4.  Find  the  greatest  common  divisor  of  x^ — y^  and  x^  —  y^, 

Ans.  X  -\-  y, 

5.  Find   the  greatest  common   divisor  of  x"^  —  IG   and  x^  —  x  — 

20  ;  x^  +  15.x  +  5G  and  x^  -f  5a;  —  14. 

6.  Find  the   greatest   common   divisor  of  x^  -\- 2x  —  3  and   a?'  -|~ 

5^_{_6;  x^—y^  and  x^-f-ic^  +  y- 

7.  Find  the  greatest  common  divisor  of  x^  —  x  —  6  and  x^  —  4; 

.-ts  — 25  and  x^  —  2x  — 15. 

§3.  The  above  rule  supposes  that  the  quantities  may  be  fac- 
tored by  some  one  of  the  methods  already  explained.  The  ge?i- 
eral  rule  depends  on  the  following  principle : 

The  greatest  common  divisor  of  two  quantities  is  the  same  as  thai 
of  the  less  and  the  remainder,  after  dividing  the  greater  quantity  hy 
the  less.     (Vide  Def.  26,  1.) 

Let  7n  and  n  be  two  quantities,  q  their  integral  quotient,  and 
?'  the  remainder,  and  let  d  be  tlie  greatest  common  divisor  of  m 
and  n,  we  are  to  show  that  d  is  the  greatest  common  divisor  of 
n  and  r. 

From  the  theory  of  division  we  have 

m  =  7iq  -\-  r. 
Now  since  d  divides  7?^   it   must   divide   nq-]-r;  and  since  d  di- 
vides n  it  must  divide  nq,  and  tlierefore  it  divides  r;  that  is,  d, 
the  greatest  common  divisor  of  m  and  n,  is  the  greatest  common 
divisor  of  n  and  r. 


GREATEST     COMMON     DIVISOE.  57 

84.  To  find  tlie  greatest  common  divisor  of  any  two  quantities: 
(1.)    Dicicle   one   quantity   hy   the   other,   and  the   last  divisor  hy 

the  last  remainder,  and  so  on  till  there  is  no  remainder.    The 

last  divisor  will  be  the  greatest  common  divisor. 
(2.)  Any  factor  common  to  all  the  terms  of  a  divisor  not 

found  in   each   term   of  the   corresponding   dividend  must 

be  rejected. 
(3.)  When  the  first  term   of  a  dividend  is  not   divisible  by 

the  first  term  of  a  divisor,  multiply  this  dividend  by  any 

quantity  that  will  make  its  first  term  exactly  divisible  by 

the  first  term  of  the  divisor. 
(4.)  The  terms  of  the  quotient  need  not  be  retained. 
(5.)    All   the   signs  of  any  dividend   may  be    at   any   time 

changed. 

EXAIVIPLES. 

1.    Find   the   greatest   common   divisor   of  x^  —  27x'  -j-  22x^  -f- 
192x  — 288  and  5x*  — Sla:^ -j- 44.t. -f  192. 

Operation, 
x^   —  27x'  4-  22x^  +  192j;  —  288  bx^  —  81a;^  -\-     Ux  -f    192 


5 

135x^4- 
81x'-f 

I10x^+960:c— 1440 
4-1x^4-1920: 

-6)- 

54x'  + 

66x^4-7680:  — 1440 

9a;^  — 
9z'-f 

11x^—128x4-  240 
9x^  —  1  OSx 

— 

20x^—  20x+  240 
20x^—  20x-h  240 

45x''  — 
4  ox  — 

729x^ 
■  5ox 

4-  396x  4- 
—  640x^  -f 

1728 
1200X 

c  3 

oox  — 
9 

•  89x^ 

—  804x  4- 

1728 

495x^  — 
495x^  — 

SOlx^ 
605x^ 

—  7236x4-15552 

—  7040X+  13200 

—  196)  — 

■196x^ 

—  196X  + 

2352 

0  x'-f      X     —    12 


Hence  the  greatest  common  divisor  is  (x -\- 4:^  (x  —  3). 

2.  Find  the  greatest  common  divisor  of  x^  -\-  bx^  4~  '''^  +  3  and 
a^s  _|_  3.^2  _^_2.  Ans.  x^  4-  4a;  4-  3. 


58  LEAST      COMMON     MULTIPLE. 

3.  Find   the  greatest  common   divisor  of  x^  -^  x^  -\-  2x  —  4   and 

x^  —  8.  Ans.  x^  -f  2x  +  4. 

4.  Find    the  greatest   common    divisor  of  x^  —  8x^  -j-  21a;  —  30 

and  x"^  —  3x^  -j-  7x^  —  3x  -\-6.  Ans.  x^  —  3x-{-  6. 

5.  Find  the  greatest  common  divisor  of  3x'  -}~  ^'^^  -\-  Sx  -\-  ^  and 

2x*  -\-  2x^  —  5x-  —  7aj  —  7.  Ans.  x^  -^x-\-l. 

6.  Find  the  greatest  common  divisor  of  x^  —  5.x^  -Y-  Qx^  —  7x-{- 

2  and  4x*  —  ISx^  +  18x  — 7  Ans.  (x  —  iy, 

7.  Find  the  greatest  common  divisor  of  a;^  —  10x^-\-37x^  —  60a; 

+  36  and  4cc'  —  SOx^  -\-  74a:  —  60.     Ans.  (x  —  3)  (x  —  2). 

8.  Find  the  greatest   common  divisor  of  x^  —  lOx^ -[- 33x  —  36 

and  3x2  —  20x  +  33. 

9.  Find   the  greatest  common   divisor  of  x^  —  13x^  -J-  56x  —  80 

and  3^2— 26x  +  56. 


85.     LEAST    COMMON    MULTIPLE. 

The  Least  Common  Multiple  of  two  or  more  quantities  is  the 
smallest  quantity  that  can  be  divided  by  each  of  them  w^ilhout  a 
remainder.     Thus,  xi/~  is  the  least  common  multiple  of  x?/  and  y^. 

To  find  the  least  common  multiple  of  two  or  more  quantities 
when  they  can  be  factored: 

Find    tJie  2)^^ocluct   of  the   smallest   collection   of  factors    ivhich 
includes  the  factors  of  each  of  the  given  quantities. 

EXAMPLES. 

1.  Find  the  least  common  multiple  of  14x2 y  and  2'8xy^z^, 

Ans.  28x2^^2, 

2.  Find  the  least  common  multiple  of  4:X^y,  6x^7/^  and  Sx^y*. 

Ami.   GO.i^f/^. 

3.  Find  the  least  common  multiple  of  x^ — 7/2^  ^ — ^  .,;,,!  x -{- y, 

-f 


GEN  j:  r.  A  L     Pv  E  Y  I  E  w .  59 

j 

4.  Find  the  least  common  multiple  of  ax  —  hx,  ay — l)y  and  x>f.  i 

Ans.  axy^  —  ^•'^^^'  I 

5.  Find    the    least    common    multiple    of  a  —  h,   a^  —  h^,    a -\-  h 

and  X.  Ans.  a'^x  —  h'^x. 

6.  Find   the   least   common   multiple   of  2a'  —  26^,   Sa^  -\- 'd>ah  -{■  i 

3^^  and  5.T.  Ans.  SOa^a;  —  306'.c.  j 

86.    To    find    the    least    common    multiple    of   two    quantities  ; 

which  cannot  be  readily  factored :  j 

Find  their  greatest  divisor;    divide  one   of  tue   quantities   hj   it, 
and  multiply  the  other  ly  the  quotient,  , 

i 

EXAMPLES.  I 

1.  Find  the  least  common,  multiple  of  x^  -j-  ^'^  -\-  2.x  —  4   and  x^ 

—  8.     {Vide  84,  ex.  3.)  Ans.  x'  —  x'  —  Sx  +  8. 

2.  Find  the  least  common  multiple  of  x^  —  8.x^  +  21.t  —  30  and 

X*  —  3x'  +  ^^^  —  3x4-  6. 

Ans.  x'  —  Sx'  +  22x3  —  38^2  +  21x  —  30. 

3.  Find  the   least  common   multiple  of  3x^  -f  Sx^  -{-  8x  -}-  d   and 

2x*  +  2x'  —  5^2  —  7x  —  7. 

Ans.  Gx^  +  ISx''  —  ox^  —  46^2  —  56x  —  35. 

4.  Find  the  least  common  multiple  of  x''  —  ox^  -f-  Ox^  —  Tx  -f  2 

and  4x3  _  i5^^2  _l  I8x  —  7. 

Ans.  4x'  —  27x^  +  71x3  _  91^2  ^  57^  _  ^4^ 


GENERAL    REVIEW. 


8f.    The  pupil  should   not  be  allowed    to  proceed  to   the  sub- 
ject of  the  fractions  till  he  can  easily  manage  the  following 

EXAMPLES. 

1.  Add  ax  to  Ix.  Ans.   (ciA-h^x. 

2.  Add  x^  -f  xy  to  y^  -\-  xy.  Ans.   (x  -f  yy. 


60  GENERAL      EEVIEW. 

3.  Add  (x -{- 7/y  to  —1}f  —  'lxy,  Am,  0^  +  3/)  (•'^  —  3/)« 

4.  Add  x^  —  lOx^  —  20  to  3x3  +  12. 

Am.  {x  +  1)  (x"  —  x^X){x  —  2)  (.X-  +  2x  +  4.) 

5.  From  1x^  +  4x7/  —  5^/^  take  x^  -f  6xj/  —  63^2.       ^7js.  (x  —  iff. 

6.  From  the  sum  of  2a  —  2x4-^?  3a  —  3x4-  2?/  and  5a  —  5x 

—  y  take  4a  —  4x  +  y  —  2.  ^7zs.  6  (a  —  x)  -}-  2/  -f  2. 

7.  Find  the  product  of  a^  +  3  (x  +  1)  by  x  —  1. 

-4«s.  a^x  —  o?  -\-  o  {x?  —  1). 

8.  Multiply  3a'  +  3a^x  by  a  —  x.  Am.  ?>a?  {ct?  —  x^). 

9.  Multiply  x^  +  hx  by  x  -\-  h,  Ans.  x  (x  -j-  ^)^. 

10.  Multiply  x2  4-  x^y^  -\-  if  by  x^  —  if, 

1  1 

Ans,.  X  —  y  -\-  xy^  —  x'^y, 

11.  Divide  1  4-  2x  by  1  —  x  —  x^ 

Ans.  1  4-  3x  4-  4x2  4-  7^3  _^  11^4  ^^^^ 

12.  Divide  x  by  1  4-  ^  +  ^'^' 

Ans.  X  (1  —  X  4-  ^^  —  x"*  4-  •'^^  —  x^  4"  ^^  *^c. 

13.  Divide  1  4-  x  by  1  —  2x  4-  x^. 

Ans.  1  4-  3x  4-  5x2  _j.  7.^3  _|.  9^^^  ^^^ 
1    i  i_  1 

14.  -Divide  x^  -\-  xy  -\-  y-  by  x  —  x^?/^  _j_  y^       Ans.  x  4-  x^y^  -{■  y. 

15.  Find  the  factors  of  x^y  4-  2x'?/2  4-  x^'.        J.ws.  xy  (x  4-  5^)^ 

16.  Find  the  factors  of  x^y  —  xj/'.         Ans.  xy  (x  -\-  y)  (x  —  ?/), 

17.  Find  the  factors  of  x^y  —  xy^. 

Ans.   xy  (x^  4-  t/S)  (x.  -{- 7/)  (x  ^  y). 

18.  From   (a  4-  ly  take  (a  —  Z>)^  ^77s.  4a6. 

19.  From  x2  (1  4-  .x2  4-  x")  (x^  —  1)    take   x2  (x'  4-  1)  (*'  —  1). 

Ans.  0. 

20.  From    (x*  4-  324)  h-  (x2  4-  Gx  4-  18)    take   (x  —  3)  (x  —  2). 

Ans.   —  X  4-  12. 
Verify   all    the  above    examples   when    x  =  9,   y  =  4,    a  =  3, 
6  =  1,  i.  e.  insert   these   numbers   in    the   given   example   and   re- 
duce ;    then  insert   the   numbers   in  the   answers,  reduce,  and  sec 
if  the  results  ajfree. 


CHAPTEK    IV. 


TE  ACTIONS. 

8S.  An  Algebraic  Fraction  represents  tbe  quotient  of  one 
quantity  divided  by  another.  Thus,  -^  is  a  fraction,  of  which  a 
is  the  numerator  and  5  the  denominator, 

(1.)  An  entire  quantity  is  one  not  involving  a  fraction. 

(2.)  A  mixed  quantity  is  one  uniting  an  entire  and  a  fractional 
quantity.     Thus,  cc  -f  |  is  a  mixed  quantity. 

(3.)  A  complex  fraction  is  one  whose  numerator  or  denomina- 

a  +  6 

tor  contains  a  fraction.     Thus,   -— -    and  -i=—  are  complex  frac- 


z  +  w 


X 

tions,  and  also   y . 


89.  All  the  propositions  in  arithmetic  in  relation  to  fractions 
are  applicable  to  algebraic   fractions. 

(1.)  The  value  of  a  fraction  is  not  changed  by  multiplying 
or  dividing  both  terms  by  the  same  quantity. 

(2.)  The  value  of  a  fraction  is  multiplied  when  the  numerator 
is  multiplied  or  denominator  divided. 

(3.)  The  value  of  a  fraction  is  divided  when  the  numerator 
is  divided  or  denominator  multiplied. 

Also  tlie  following : 

(4.)   The   value  of  a  fraction  is  not  changed  by  changing  all 


— X' 


the    sijrns.      Thus,    —  =  =  x,    and    —  =  —  =  —  x,    and 

o  ■        X  — X  '  X  — X  ' 

„o  o  2  o 

35'  —  y-         y  — X-  , 
= r=z  X  -\-  y. 


62  F  E  A  C  T  I  0  N  S  . 

(5.)  When  ii  sign  is  placed  before  a  ffiictlon  with  a  poly- 
nomial numerator,  the  sign  does  not  belong  to  the  first  term  of 
the  numerator  but  to  the  whole  fraction.  Observe,  in  reducing, 
the  rules  of  addition  and  subtraction. 

,,,,  ,     —  a-\-b  —  c  —  a-{-h  —  c        ,      .  —  a-\-b  —  c  a  —  fc  +  c 

ihus,   A -. = z :    but =  — ~ — ■, 

'      '  0  5  '  6  5 

90.   (i.)   To  reduce  a  fraction  to  its  lowest  terms: 

Divide  the  numerator  and  denominator  hj  their  greatest  common 
divisor,  or  cancel  the  factors  common  to  the  riumerator  and 
denominator. 

EXAMPLES. 

1  i  OC'  V  m  or, 

1.  Eeduce  -— —  to  its  lowest  terms.  An^.  -— . 

85.T  y  m  5?/ 

2.  Eeduce  -—: — r^,   and  — — to  their  lowest  terms. 

Ailxifz^      xyz  \\)'i)xyz 

S.'z:^  -\-  7ax        .      -  .        5.T.  -f-  7a 

3.  lleduce  — ,    _    „    to  its  lowest  terms.  Ans.  . 

ox  -[-  5^2  3  -[-  Dx 

Sx^  -\-  lC)x?y  x^  —  ?/^  x^  —  y"^ 

24:X^  -j-  i^2x^y '  x'^  -j-  2xy  -{-•  y^  x^  —  2xy  -j-  y^ 

a;2  _  9x  -I-  20         .      ,  ^       x  —  5 

5.  Eeduce  —^ -—-  to  its  lowest  terms.  Ans.   — — -  . 

x^  —  x  —  12  X  -{-  o 

a'2_2.x  — B5     x^  4- l^x -4- bQ        ,        x""  —  IQ 

6.  Reduce     ,   ,   ^ — r-.-^ »  -^^-r t-t  ^^^ 


ay^  _|_  8a;  H-  15  '    x^ -{-  bx  —  li  x^  —  x  —  20' 

x^y  —  xy^  X?  —  ?/'  o?  —  y^ 

7.  Reduce  — 0021 T'  ~4~^ — 2  2    ,    "4  ^"^^  ~1 ^4 

X  y  —  Zx^y^  -j-  xy*      x^  -j-  X'^y^  -\-  y^  x^  —  y 

27x^  —  04^3  3,^  _  4^ 

^-  1^^'^"^^    SLx^  -1-  l^^y''  -I-  2563r^  *     ^  "^'   9a;2  _  i2.t3^  -f  16/  ' 

a?  —  4.x  x'  4-  2.^^  _  8.T:  —  16  a;^  -f-  y\ 


0.  Reduce 


.^3  _]_  6a;2  _j_  8x '  3  (r/j2  —  4)  '  x'  -|-  x^'y''  -f-  3/ 


a;3  _  8r«2  _L  21.r  —  30  ,         r^;  —  5 


FRACTIONS.  63 


11.  Reduce    .^   .    ,    ^  „ — —  and 


2ic''  -j-  2x^  —  5x^  —  7x  —  7  rzj^  -j-  ox^  —  x  —  3  ' 

(Vide  §4.) 

(ii.)  To  reduce  a  fraction  to  an  entire  or  a  mixed   quantity: 

Divide  the  numerator  hj  the  denominator,  ivriting  the  remainder^ 
if  there  he  any,  icith  the  denominator  under  it,  at  tJie  right 
of  the  quotient,  ivith  its  sign  i^rejixed. 

EXABIPLES. 

x2x  X 

1.  Keduce  — —  to  a  mixed  quantity.  Ans.  x  -| . 

—  2i7x  X 

2.  Reduce  — ^-^ —  to  a  mixed  quantity.  Ans.  —  2x  —  -— -. 


• 

13 

x'-i- 

ax 

X 

x" 

— 

«= 

3.  Reduce  to  an  entire  quantity.  Ans.  x  -f-  a. 


~        -  .  .  a* 

4.  Reduce  — — to  a  mixed  quantity.  Ans.  x . 

X  X 

K    ^  .        2)5  — 9CC+20  .     ^  .         ^       ^  8 

5.  Reduce  — z .-  to  a  mixed  quantity.    Ans.  1 . 

ic'  — X  — 12  ^  ^  .-c  +  3 

{Vide  90,  (1),  5.) 

Sx^*  —  Qa^      5x2    1    10^2  5^3  _|_  7^^ 

6.  Reduce ,  =- and  -— -r ^r — 

6x  ox  5x'  -f-  3x 

x^  —  y^     x^  —  y^     x^  -4-  y^  x^  —  v^ 

7.  Reduce  ^,  ^,  — -^-^-  and ^. 

X  —y       x  —  y        x-^y  ^ -\- y 

% 

x^  ^  xy  +  y^  x^  +  x^y  +  xy^  +  y^ 
*8.  Reduce             i    i           and  x  +  v 

X  -\-  x^y^  -\-  y  "T*  ^ 

^3  __  4^,  ^3  _^  g^2  ^  8x       ,  x^  —  2x  —  35 

9-  I^educe  -3  +  a^2  +  8x'         x3-4a;         "^^  ^+-8^+15' 

X^  +7/2  1  4-  w' 

10.  Reduce ^-   to  a  mixed  quantity.    Ans.  x  —  1  4- ~, 

x  -\-  1  ''  1  -f  a; 

^  ^     _   ,         1  +  V^     x^  —  y"^      1  —  y"^  1  —  X 

11.  Reduce    -p^  ,  —^,  ^  and  — --. 

1  -f  ^-       2-'  —  1        ^  —  1  \  -\-  X 


64  F  E  A  C  T  1  0  N  S  . 


12.  Iteduce and 


u> 


;  +  y  x—y 


13.  Keduce  --- — ,  — - —  and  — .-  ,    ^  , rr* 

y  -^  X     X  -{-  y  x^  -\-  6x^  —  x  —  6 

x^  —  Sx^  +  21cc  —  30  x~b 

(ill.)   To  reduce  an   entire  or  a   mixed  quantity  to   the  form 
of  a  fraction : 

(1.)    Multiply  the  entire   quantity  by  the  j^^^osed  denominator, 

and  the  2^roduci  will  he  the  numerator ;  or, 
(2.)    Multiply   the   entire  ^xw^  by  the  denominator  of  the  frac- 
tional 2)art,  and  add  or  subtract  the  numerator,  according  to 
the  sign  before  the  fractional  part,  then  place  the  result  over 
the  given  denominator. 

EXAMPLES. 

1.  Reduce  x  to  a  fraction  whose  denominator  is  1,  2,  or  7. 

X      2x     7x 

X  ,      /.  p       -.       .  4        12a; 

2.  Reduce  x -{-  ~—  to  the  form  oi  a  fraction.  A7is.  -— - 

X  1 0^ 

3.  Reduce  x  —  — —  to  the  form  of  a  fraction.  Ans.  — — 

>.    T,  ^  ,1         1  1  .       llri;+l       ^   llfl^-l 

4.  Reduce  x  -{-  —-  and  a;  —  — - .       Ans. — and  — 

a  —^  X 

5.  Reduce  a  -{-  x -{ .      Vide  90,  (i.)  Ans.  2  (a  -f  a-) 

a^  4-  a;2  a^  -f  x^  x^  4-  iP 

6.  Reduce  a  —  x  -{ ; ,  a  —  x — ,  x  -{•  y 


a  -\-  x  a  -\-  X  ^  -\-  y 

2xy                   ,   a;^  +  V^       -,  2a;?/ 

7.  Reduce  x  -\-  y ; — ,  x—y-\ and  x  —  y  -{■ 


x-\-y  x—y  x—y 

a^     I   ?/  a^  ——  y^ 

8,  Reduce  x^  +  xy  +  :?/^  +  - — — ~  ^"^  ^^  ~  xy  -f  y^  +  * — - — . 

x—y  x-\-y 


FRACTIONS.  G5 


X 


f  o  ...,.,3  •'«'+^^ 


:2  ,,2 


9.  Reduce  x^  —  crij  +  f—-    ,         ,     ,,    .x=  — .r^/ H-^/'- 


a^'+a^^+y^  "  -^^-^^iz+i/ 


1  +  ^     ......       1  + 


.3 


10.  Eeduce  1  —  x  —  ^    ,        ,   1  -f  a;  +  .t^ ^ and 

1  +  a;  1  —  X 

1  +  re  +  x^ ^t^ 


1  —  x 


1  +  rt  +  X  «■''  —  x^  , 

11.  Reduce  tt  +  1 ,  a  -^  x , — ^  and 

^2 ][g  (5^ 29 

12.  Reduce  5  +  -^_     _,^^  .  (F/t?e  90,  (i),  ex.  6.)  ^«s.  -^^^^  - 

13.  Reduce  1 ^-^ — ^Vr  ^^^^  ^  H ^ . . 

x^  -{■  ax  —  14  x^  —  4x 

(iv.)    To    reduce   fractions    having    different   denominators    to 
equivalent  fractions  having  a  least  common  denominator: 

(1.)  Reduce  the  fractions  to  their  lowest  terms,  unless  they  are 

so  given  as  to  admit  of  no  reduction. 
(2.)  F'ind  the  least  common  multiple  of  all  the  denominators. 
(3.)  Divide  this  multiple  hj  the  denominator  of  the  first  reduced 

fraction,    and   multiply    the    quotient   hj    the    numerator,  and. 

lurite  the  product  over  the  mtdtiple.      Do   the  same  for  all 

the    fractions,    and   the    resulting    fractions    will    be    tho?e 

required, 

EXAMPLES. 

X  2ax  ^   X  -\-  x"^  .     -,     ,   r 

1.  Reduce  -^- ,  — r  and  — to  equivalent  fractions 

x^  -j-  xy     X  —  xy^  X  —  xy 

having  a  least  common  denominator. 

Solution. 
X  2ax  X  -\-  x^ 


x^  -\-  xy^    x^  —  o:y^  '    x^  —  xy 
1  2a  \-\-x 


y      x'—y^      x—y 

D 


=  fractions  given. 
=  fractions  reduced. 


66  r  R  A  C  T  I  O  N  s . 

x  —  y           2a          X  +  x^  -\- y  +  ^y         ^     ' .  .     , 
,    — -„ ,    ~ r =  tractions  required. 

^  —  y      ^  —  y  ^  —  y 

■       ^    rc2  4-  4.-^  +  3  x2  +  8.x  +  15 

2.  Eeduce  x  +  10,  — r  and  — — to  equivalent 

x^  -\-  x  —  ij  a;^  —  25 

fractions  having  a  least  common  denominator. 

Solution. 

a;  +  10      fc2  +  4.T.  +  3  ic^  +  8.T  +  15  ^       . 

-T-'     x^  +  x-^^        -^^325—        =  f-ctions  given. 

^±25,  (^+3)_^l)      (^  +  5)0^+3)  ^ 
1       '  (.c  4-  3)  {x  -  2)  '    (x  +  5)  (a)  -  5) 

aj+lO  rc+1  cc  +  3 


X  —  2  '  £0  —  5 


=  fractions  reduced. 


^3  +  3x2  — 60x+ 100      x2  — 4a;  — 5_  x''-\-x—Q     fractions 

^_7x4-  10         '    a;2  _  7a;  +  10  '  x^^^^Tx+TO    required. 

x     a?           a;  Qx      4x            3x 

3.  Reduce  —  ,  —  and  -  .  Ans.  — - ,  — —  and  — r  . 

jj      o             4  ij       1-j              ii^ 

11,1  .         ?/,2       a;^         _    xy 

4.  Reduce  -  ,  -  and  -  .  Ans.  - —  ,  • and  . 

X    y           z  xyz     xyz           xyz 

11                   1  X^       X                 1 

5.  Reduce  -,  -^  and  —  •  Ans,  '—,  -^  and  -z. 

X       XT                X^  X"*       X*                X* 


(v.)  To  add  fractional  quantities  together: 

(1.)   Reduce  the  fractions  to  the  [east  common  denominator. 
(2.)  Add  the  numerators,  placing  the  sum  over  the  least  common 

denominator. 
In    mixed    quantities,   add    the   entire    parts   first,   to  which 

annex  the  sum  of  the  fractional  parts  by  the  proper  sign, 

EXAMPLES. 

1.  Add  the  quantities  x  -\ ; — ,  3a; ,   —2x-\ . 

X  -\-  y  x^  —  y^  X  —  y 


FRACTIONS.  67 

Sohdion. 
The  sura  of  the  entire  parts  is  re  +  3a;  —  2a;  =  2x. 

=  fractions  given. 


ic  +  y '        x^  —  if        x—y 

^""^„ ,     -^ — -„ ,        4 —1       =  fractions  reduced,   (iv.) 

a^  —  ?/''       ic  —  y^         re  —  ?/ 

2^ 1 

2rc  -J — :        =  the  sum  as  above. 

X  —  y 

.  XXX  T*^         i  i  iX  \-  i  X       . 

2.  Add  the  fractions  «  ?  o  >  7  and  -.    JLns.  -—  =  re  +  — — .  (ii.) 

or  or  ^  QC/ 

3.  Find  the  sum  ofrrr-  +  ^r-7+-^+-H-r. 

11  \6  LKi  OJ 

.11,      .111,      .1     1 

4.  Find  the  sum  of  -  +  -  ,  also  of  -  -\ 1 — ,  also  oi  — h  -  o  • 

X      y  X      y       z  X      XT 

.  1        1       ,        .    2  3  5 

5.  Find  the  sum  of  -=  H — r ,  also  of  — — | ^  +  -^— _ . 

X        x^  X  y       xy^      x^y 

rc+1         ,  CC+  1  ,         (x-^\f 

6.  Find  the  sum  of  5—  and  r— .  Ans.  r-^* 

12  3 

7.  Add  the  fractions  — ■ — ,  -r ~  and  . 

X  -\-  y    x^  —  y^  X  —  y 

2(2rc  +  3/  +  *l) 


A71S, 


,.2      -,,» 


Q      1  ^  +  y  1  ""^y 

^  —  y      XT  -\-  '^y  -\-  y      ^  —  ^ 

1  1  11 

'  x'^  —  y^       rc^  4"  x^y  -\-  ^y^  -{"  y^       x^  -^  y^       x^  —  y^ ' 
1  1 

10. n —  + n — • 

re  4-  x-y^  -j-  y        x  —  rc^z/^  -|-  y 
^^        4re       ,    re  +  2  7re  +  26 


X  —  2^x  +  3'  *       'rc'  +  .T  —  6 

19    ^  +  ^  _L  ^-'^ 
x  —  7        X  +  o 

13.  ^±t  +  -li^. 

re  —  D        re  —  4 


68  F  K  A  C  T  I  0  N  S 


14,  __1 ^ —  _| :• J 

X-  -\-  X  —  6  x"^  —  25 

,^  a;2— 1  cc2  — 36 

15.  -^— -^-  + 


;t2  _|.  (j,^  _  7  ^  ./;2  _  2.^  _  48  • 
x^  +  Sx  4-  "i       X-  -\-  Ax  —  5 

1(^.     -.  o  .     + 


^  ^    a;'  +  a;^  4-  cc  4-  1  ^'  —  ^^  —  ic  4-  1 

17. ^ ' 1 ■ —  . 

x^  -\-  x^  —  X  —  1  x^  —  x^  -\-  X  —  1 

^^    X  4- y        X  —  y  ^-      x  —  y  x  4- y 

18.  —LA  +  __^ .  19.      — ^  4  _^iL 

r.r.     X  —  y         X  A-  y                                    ^.        2x  1 

20.  -T— ^  +  -rTA.  21., -.4- 


X*  —  ?/*        x^  -\-  y^'  *   1  —  a;^       a;  +  1  * 

2  3  4 

22.  ~r ^^l  +  T— r^  + 


^  X  y 

23. ^^ — — -^  4- 


(14  a))  (cc4-y)       (1-y)  0-^  +  3^)* 
24  _1_  +  — L_  +  _JL_. 

4  (1  ^  ic)    ^   4  (1  ~  :z^)  ^  2  (1  -  ic2) 
25.  Add  7.T  +  I  ^  51  to  ^  4-  7y  -  99. 
^o     .  ,,   3aj        2v         61  2x        3y         9 

^«..  to  last  -g?  +  ^^  -  g-. 

(vi.)  To  subtract  one  fraction  from  another: 

(1.)  Reduce  the  fractions  to  the  least  common  denominato7\ 
(2.)  Subtract  the  numerator  of  the  subtrahend  from  that  of  the 
minuend,  l^lacing  the  difference  over  the  least  common  denom- 
inator. 
Mixed  quantities  may  be  reduced   to   a  fractional  form,  or 
tlic  parts  subtracted  separately. 


FRACTIONS.  69 


EXAMPLES. 

1.  From  3x  -j-  ):   take  x —  . 

Ojyej'ation. 

X  4:X  ... 

3x  -\ — .  X ^  .        =  quantities  given. 

Tx  on 

-— .  -.  =  quantities  reduced,   (ii.j 

A  0  . 

35x       2x       33x        _  ox 

Then zr-  =  -^rr-  =  ox  -\-  -— »     Ans. 

10         10         10  ^  10 

,^2  _^  6a;  4.  8     ^         0)2  +  3a;  —  40 
x^  —  X  —  6     '         x^  -{•  X  —  30  * 

Operation. 


(a;+2)(x  +  4)  (x  +  8)  (x  — 5) 

(a;  +  2)  (a:  —  3)  (x  +  6)  (a;  —  5) 

05+4  a;+8 

X  —  3  a;  -j-  6 

^2  _|_  lOx  +  24  a:'  4-  5a;  —  24 

a;2-j-    3x  — 18  x'-fSa;  — 18 


=  fractions  factored. 
=  factors  cancelled. 

=  fractions  reduced,   (iv.) 

5a; +  48 
a;^-j-  oa;  — 18 

\  \  y X  1  1 

3.  From   -  take  -.  Ans. .      4.  From  3a;-] —  take  a;-!--, 

X  y  xy  X  y 

4x  11 

5.  From  7x  take  -rr-.  6.  From  -   take  -^. 

5  X  x^ 

7.  From  — +r  tiike .  8.  From  — ^-^  take  — -^  . 

X  —  1  x-\-l  X — y  ^-rV 

y.  Irom  — !— r:  take 


X  —  6  a;  -|-  3  * 

1^    ,  .a;2_|_4^_12  x2-f2a;  — 15 

10.  I  rem     „   ,    , Y7  t'^kc  —7-7—^ L-5r  • 

a;^-fOa;  —  14  x^ -{- ox  —  18 


70  F  R  A  C  T  I  0  N  S  . 


11.  Prom  ^^^^^  take  -^^ZTsS"  * 

12.  From  -^^3  take  ii=^.     13.  Frdm  ^^  take  4=4. 

•"^  "1^  cc  — ?/''  re  — i  X  -\- I 

14.  From  — ^  take »— -^J^. 

ic — 3/  x^ — y^ 

^  „    _  1  ,  X  —  2 

15.  Irom  — — -  take 


X  -\-l  o?  —  x-\-\' 

16.  rrom  -  +  -  take .  An^,  -. 

y       z  y       z  z 

,  ^    __  ,      hm-\-  c 

17.  I*  rom  711  take  — ^  . 

a-\-  0 

-.  r.    ^.  a6  4-  <^c  4-  ^c       ,      11        .1  •    .     , 

18.  I  rom --—, take  -,  -  and  -   respectively. 

19.  From  263  _i±^  take  h'  —  l^. 

3  4 

11a:  — l(f    ,      „     .   3a:  — 5 

20.  From  3a:  -] — —    take  2a:  -\ — . 

JLO  • 

(vn.)  To  multiply  fractional  quantities: 

(1 .)  Reduce  mixed  quantities  to  the  form  of  a  fraction,  (iii.) 
(2.)   Factor  each  numerator  and  denominator,  and  cancel  such 

factor's  as  are  found  in  hath  numerator  and  denominator. 
(3.)   Multiply   the   remaining  factors   of  the  num.eratoj'S  for  a 
new  ?iumeraio7\  and  those  of  the  denominator  for  a  new  de- 
nominator. 
(4.)   Reduce  the  result,  if  necessai-y,  to  a  mixed  quantity.  (11.) 


1.  Multiply  a  A bv 

a  —  X     "     X  -'-  X 


EXAJIPLES. 

2  or.2 

2"* 


FRACTIONS.  71 

Operation, 

ffn^  fl^  ^^ 

a  A .         ^-_  .  =  given  quantities. 

a  —  X  x~\-  x^ 

a?  (a  4-  x^  (a —  x)  .  -       « 

By  II.       ■ .         -^-~ — r — -  .         =  quantities  lactorea. 

a  —  X  X  [1  -\-  X) 

Tiien  — ^ r- .  =  result  required. 

X  (1  -{-x) 

Or*^  qj^  rf*   2/ 

^  "^    X?  —  2,xy  -\-y^    "'    x^  -\-xy' 


Operation, 


x'^  —  y^  X  —  y 


=  given  quantities. 
=  quantities  factored. 


x^  —  2xy^y-  x" -\- xy 

(•^ —y)Q^—y)        '  ^  (^  H-  y) 

x^  +  y^  ,  y^                         .           .    , 

Then                     —^^  =  x -\-  '—  =  quantity  required. 

X  X 

a?bx  ,       Q>m^n^  .        2mn 

3.  Multiply  -^^^  by  -;jj^  .  Ans.  — -. 


3mn  a^  bx 

^     x-\-l   .      x  —  1 


oj- 


4.  Multiply  — —-  by  .  An^,  1, 

X -L  X  — 1~  X 

5.  Multiply  2  +  -^  by  2  —     ^ 


X  —  1  x-\-\ 

^'  2  + T  by  — -r  7.     3  ,  ^    '^,     ,  by  -^H:^. 


^'-2''        by  4^.  9.-^^,l>y        ''+^ 


a^  —  2xy-\-y'^         x^-^-y"^'  '  x?  —  ?/^         .r^  —  ^y-\~y 

^^    x^  —  y^  x^4-y^  a;  +  V 

x^  —  3/^       X.  -\-  xy  -\-  y^        x^  —  xy  -\-  y' 

'      x'  —  x—    2         rc^-f  a;  — 20 

11. X   '■— = 

x^  —  X  —  12        x^  —  6.T  —  7 


2* 


72  I'  I>-  ACTIONS. 

\         x^-{-2ax  —  'Sa^J  \         x^ -{- ax  —  6a*/* 

16.      1+-^11-      X      1 -—). 

\         a  —  X  J  \         a-\-x  I 

18.      a  — .T-] 4^—      X    [a  —  x 4--    • 

20.  C^'  +  x  +  l)  X  (i-l  +  l). 

21.  (.  +  1  +  1)   (.x-1  +  1). 


22.  (a  +  i^)  X  -"-t-^  =  c  +  t7. 
a  -T-  6» 


(viii  ^   To  divifle  fractional  quantities : 

(1.)    'Reduce  mixed  quantities  to  the  form  of  a  fraction, 
(2.)  Invert  the  divisor^  and  then  j^roceed  as  in  multiplication, 

EXAMPLES. 

1  -4-  X*  \  A-x^ 

1.  Divide  l—x  —  --^   by  1  +  a;  —  re"  —  ^  ^  '    . 

1  -f-  ^  1  —  ^ 


FRACTIONS.  73 


Operatioju 

1  —  a;         1  —  X 
„,,        —  2cc2         1  —  X        1  —  X     . 

Then  nr^  X  iri:^^  =  np^  ^»'- 

7ic?/^         4:Xv^z  21 

2.  Divide  —^  by  -4--  ^^^s-  T^a- 


2^  +  3/     .     ^+3^ 


'S- 


4. 


a; — y  '^ 


5.   (a  — a;  +  — ^)  --     a  — a; ]— -). 

\  a-j-x/  \  a-{-  X  / 


a;2_4      ^    a;2  — 9a^  +  14  -    a;2_4 

1_^2  l4-3x4-2a;2  1  — 3x  +  2a^ 

^-  1  _|.  5a;  _|_  Ga^^ -^    l_j_.^_6x2*  ^^-  fipi^qTi^a- 

a^  — x^  ^    a  — a;  2,2 

a^  -|-  2aa;  -\- x^         a -{- x 

11.    '''  +  ^^y  +  f  ^    ,    ^  +  y         _  J«..  X  -y. 

cc^  —  ;y2  X  —  2x1/  -j-  ^2 

'^■('+jqn)*('-ST)- 

7 


74 


FRACTIONS 


( 


14.    [l  —  x 


1  -i-x 


■)-( 


l-~x 


l+x'] 


) 


15. 

16. 
17. 

18. 


:.(, 


1  -{-  X  /     '     \  1-j-x 

/  l-\-x^\  /,  l-l-a;2\ 

2xy 


x^J^xy^-y 


x  +  y 


x  +  y 


)*{ 


x-\-y 


x-^y 


)■ 


x^^-y^' 


19. 

20. 

21. 

22. 

23. 

24. 

25. 

26. 

27. 

28. 


.'»— 3/ 


X -\- y }     '     \x — y        x-\-y)'^ 


>^  4-y 


r 


( 


r 

A71S.    X, 


X-^y        X  —  7j\    _^    /X^ 


—y   ,   »^H-3/ 


-  4- 

x—y   '   x-^yl  \x'—y^       x^-\-y 


■?)■ 


/x-}-!  __x  —  l\   ^      x^ 
\x  —  1       x-j-l/         ^  —  1' 

\x—y 


x^  —  2xy-\-y^\         I     1 


-<4yj5. 


a;  +  l' 


Divide  1  by 
Divide  1  by 
Divide  1  by 


x-—y^ 

ah  -\-hc  -\-  ac 
2abc 

ah  -\-  he  -\-  ac 

ah  -{-  he  -\-  ac 
2ahc 


+ 


/     *     \x -{- y    *   x^ 
1 


■y'      x  —  yj 


1 

6* 

1 

a 


(--a  -  (-')• 


Ans.  x^  A-  x  -\ 1 — - , 

X       x^ 


F  Jl  A  C  T  I  0  X  S  .  75 


REVIEW     OF    FRACTIONS. 
91.  Perform  the  operations  indicated  in  the  following 

EXAMPLES. 

X  —  1    ,    X  —  4        X  —  7    ,     llx-^bb  . 

-4«.s.  X. 
x-\-\  X  —  1  X  —  2 

Q^2  11^1      ft     "I       9^2 7.y.  J^   '-^       '      ^ 


3^2  — 11a:  4- 6     '    2x^  — 7x  +  3    '    <dx^  —  1x-{-2' 

Qx?  —  9x  +  7 


5. 


Qx^  —  25.7:2  4-  23x  —  6  • 

3x3  _|_  8a:2  -I-  8x  +  5  x»  +  Sx^  _j-  7x  +  3 


2x^  +  2x3  — 5x2  — 7x  — 7         a;'4-3x2  — X  — 3  * 


2x'4-5x2  — 5x  — 12 

"^*  "2x3  — 2x2  — 7x-j-7* 


2x  —  14  3x  —  21  4x  —  28 

C.  -T— ^^ — ^  + 


x^  —  Zx  —  2^    '    x2  — 11x4-28         x3--7x2  — 16x4- 112/ 

.  5x 


x2  — 16* 
7.  ^^  X  ^^-^^.  ^....  27  +  9^4-3/ 4-y. 

3  2x  — 15  2  12x(10x4-  3) 

^-    2^TZ  3"  ~  4^2~+~9   "■  2x-f  3  *  ''^'       IGx-'-Sl      ' 

12  2 


a;4  4-  4«^  ^  x2  +  2«x  4-  2a2        x2  —  2ax  4-  2a2  * 

,         1  —  8crx 
Vide  61,  ex.  36.     Ans.  — ___ . 

X*  4-  4a* 


CHAPTER    V. 

92.  EQUATIONS    OF   THE    FIRST   DEGEEE. 

(1.)  An  equation  is  an  algebraic  expression  showing  that  two 
quantities  are  equal.     (^Vide  Def.  15,  also  Def.  3  and  4.) 

(2.)  An  equation  is  of  the  Jirst  degi^ee  when  its  unknown 
quantity  is  involved  to  i\\Q  first  power  only.      (Vide  Def.  13.) 

(3.)  A  numerical  equation  contains  only  numbers  and  the  un- 
known quantity.     Thus,  3x  -f  5a;  =  30. 

(4.)  A  literal  equation  contains  letters  representing  known 
quantities.  Thus,  ?>x  -\-  ax=  h,  where  a  represents  5  and  b  30 
of  the  equation  in  (3.) 

(5.)  An  identical  equation  is  one  in  which  both  members  are 
alike,  or  in  which  either  member  is  the  result  of  operations 
indicated  by  the  other.  Thus,  |— ^  =  — ^ ,  and  ^1^  =  1  +  2aj 
-f  2x''  +  &c. 

(6.)  An  equation  is  verified  when  on  the  substitution  of  a 
quantity  for  x,  it  is  rendered  identical.  Thus,  if  for  x  in  the 
equation  x  -f  3x  =  24,  the  number  6  is  substituted,  it  becomes 
G  -f-  18  =  24,  where  6  is  the  only  number  which  will  render  the 
given  equation  identical. 

(7.)  The  solution  of  an  equation  consists  in  finding  the  quan- 
tity which  will  verify  it.  This  quantity  is  called  a  root  of  the 
equation. 


EQUATIONS      OF     THE     FIRST      DEGKEE.         77 

(8.)  The  solution  of  an  equation  depends  upon  one  or  more 
of  the  following  self-evident   propositions,  called  axioms. 

93.  AXIOMS. 

(1.)  Quantities  which  are  equal  to  the  same  thing  are  equal 
to  each  other. 

(2.)  If  to  equal  quantities  equal  quantities  be  added,  the  sums 
will  be  equal. 

(3.)  If  from  equal  quantities  equal  quantities  be  subtracted, 
tlie  remainders  will  be  equal. 

(4.)  If  equal  quantities  be  multiplied  by  the  same  or  equal 
quantities,  the  products  will  be  equal. 

(5.)  If  equal  quantities  be  divided  bj  the  same  or  equal  quan- 
tities, the  quotients  will   be  equal. 

(6.)  If  quantities  are  equal,  their  like  roots  are  equah  (^Vide 
Def.  13,  3.) 

(7.)  If  quantities  are  equal,  their  like  powers  are  equal.  (^Vide 
Def.  13,  2.) 

SOLUTION    OF  EQUATIONS    OF  THE  FIEST  DEGREE    CONTAINING 
ONE    UNKNOWN   QUANTITY. 

94.  The  object  of  every  change  in  equations  is,  ultimately, 
to  make  the  unknown  quantity  x  constitute  the  first  member, 
and  the  known  quantities,  reduced  to  their  simjolest  form,  the 
second  member.     The  equation  is  then  solved.       ( Vide  92,   (7.) 

9.5.  To  solve  an  equation  of  the  form  ax  =  h. 

Since  ax  =  h,  by  axiom  (5)  x  =  -  .     Hence, 
Divide  the  equation  by  the  coefficient  of  x. 

EXAMPLES. 

1.  Solve  the  equatiofi  2x  =  10. 


78        EQUATIONS     OF     THE     FIRST     DEGREE. 

Solution. 
2x^=  10  (1)  =  given  equation, 

x  =  5  (2)  =  required  equation. 

Equation  (2)  is  obtained  by  dividing  (1)  by  the  number  2, 
which  is  the  coefficient  of  x.  The  number  5  will  verify  the 
given  equation,  for  2x5  =  10. 

In  the  same  way,  solve  and  verify  the  equations 

2.  Ix  =  21.  Aiu.  X  =  3.       8.  12.x  =  156. 

3.  5x  =  25.  9.  13.x  =  169. 

4.  4.x  =  14-1.  10.  ax=a?.     An&.  x  =  a. 

5.  3x  =  15.  11.   hx  =  ah  -\-  IP.      Ans.  x=  a  ~\-  h, 

6.  10x=20.  12.  ax  =  U.     Ans.  x  =  ~ 

a 

7.  9x  =  729.  13.  2ax  =  c.     Ans.  x  =  ~. 

la 

96.  If  both  members  consist  of  several  terms ; 

Unite  these  terms  (by  33)  and  then  divide  hj  the  coefficient  of  x. 

EXAMPLES. 

14.  Solve  the  equation  2x  -\-  3.x  =  30  -f  15. 

Solution. 
2.x  +  3.x  =  30  -f  15  (1)  =  given  equation. 

5.^:  =  45  (2)        {Vide  33.) 

.X  =  9  (3)  =  (2)  --  5.     Axiom  (5.) 

2x9  +  3x9  =  30 -f  15  =  verification. 

15.  Solve  the  equation  5.x  —  2.x  =  50  —  20.  Ans.  .x  =  10. 

16.  Solve  the  equation  5.x  +  3.x  —  2.x  =50  —  40  +  2. 

Ans.  x  =  2. 

17.  Solve  the  equation  20x  —  18.x  +  4.x  =  100  —  70  +  30. 

Ans.  X  =  10. 

18.  Solve  the  equation  11a;  +  15.x  —  10.x  =  100  —  10  +  14  —  8. 

Ans.  .X  =  6. 

19.  Solve   the  equation   ax  +  hx  =  r  +  d. 


EQUATIONS     OF     THE     FIRST     DEGREE.         79 


Solution, 
ax  -]-  hx  ■=  c  -\-  d  (1)  =i  given  equation, 

(a  +  Z))  x  =  c  +  cZ  (2)   {Vide  75,  ex.  3.) 

c  -\-  d 


X 


a  -{-  b 


c  -4-  d       -        c  -{■  d  ,  .... 

Then,     a  x       ,   -.  +  ^  X  — —j  =  c  -j-  d     =  verification. 
a  -{-  0  a-\-  0 

If  a  =  2,  h  =  3,  c  =  30  and  d=lb,  then  (ex.  14)  x=9. 

If  a  =  5,  Z)  =  —  2,  c  =  50  and  d=  —20,  then  (ex.  15)  x  =  10. 

20.  Solve  the  equation  ax  -\-  hx  —  ex  =  d  —  e  +  /'. 

Ajis.  X  = . 

a  -\-  0  —  c 

If  a  =  5,  Z>  =  3,  c  =  2,  cZ  =  50,  e  =  40  and  /=  2,  tlien  x  =  2. 

If  a  =  6,  Z/  =  4,  c  =  5,  cZ  =  40,  e  =  20  and  /=  10,  then  x  =6. 

X 

QK,   To  solve  an  equation  of  the  form  -  =  Z>. 


a 


X 


Since  -  =  h,  by  axiom   (4)  x  =  ah.     Hence, 

a 

Multiply  the  equation  by  the  denominator  of  the  fraction. 

i:xa:mples. 

X 


1.   Solve   the  equation  -  =  7. 

o 


1  =  ^ 

.T  =  21 
11  =  7 


Solution. 

(1)  =  given  equation. 

(2)  =  required  equation. 
=  verification. 


Equation   (2)  is  obtained   by  multiplying    (1)   by  the  number  3. 
In  the  same  manner  solve  and  verify  the  following. 


2,  ^=20. 
5 


3.  ^  =  6. 


Ans.  X  =  100. 
Ans.  x=  9. 


4.-  =  X«. 


5. 


3x 


15. 


80.     EQUATIONS     OF     THE     FIRST     DEGREE. 


6.  ^  =  14.    Ans.x=  IS. 

7.  ^=18.    Ans.x=24, 


11.^ 
13' 


—  99 


0.^^=39. 


1A       ^^  2 

10.  — -=  a^ 

o 

11   ^^^ 

11.  =  a. 

€ 

12.^=26. 


Ans.  X  =  3(X, 


ac 

Ans.  X  =  — p 

46 


Ans.  X  = 


130( 
"36" 


13. 


X 


a-\-h 


c.    Ans.  X  =  (a  -\-  Jj)  c. 


98.  When  there  are  several   fractions : 

Multiply   the  equation  hy   the   least   cominon  multiple  of  all  the 
denominates,  after  which  proceed  as  in  96. 


EXAMPLES. 


X  X 

14.  Solve  the  equation  -  -f  -  =  10. 


X  X 


Solution. 

(1)  s=  given  equation. 

(2) 

(3)  {Vide  96. j 

(4)  =  required  equation. 

=  verification. 


?>x-\-2.x=  60 

bx  =  60 

.x  =  12 
12        12 
-2-  +  X  =  l« 

Equation   (2)  is  obtained  bj  multiplying  both  sides"  of  (1)  by 
6,  the  least  common  multiple  of  2  and  3. 

15.  bolve  the  equation  -  -f  — —  =  13. 

Z        6         '± 


Solution, 


X       4iX        ox 

Qx  -f-  16a:  —  dx=  156 
13a;  =  156 


(1)  =  given  equation. 

(2)  =  (1)  X  12. 

(3) 


EQUATIONS     OF     THE     FIEST     DEGREE.  81 

£c  =  12  (4)  =  required  equation. 

12       4x12       3  X  12       ^_  .-      . 

: —  =  13  =  verification. 

2^3  4 

In  the  same  manner  solve  the  equations, 

16    ^  +  ^  =  23.  Ans.  x  =  20. 

4  5 

17.  x  +  ^  +  -^  =  11.  Ans.  a;  =;=  6. 

2       o 

18    ?  +  '-_- =  7.  Ans,x=  12. 

'2^3       4 

^^    2x       4:X        X        ^^  .  .- 

^,     4.x        3a;        2^7        ^  ^  -  o- 

21 =  15.  ^  Ans.  X  =  3o. 

5  7    ^  35 

22.  -  4-  6x=  38.      yl»5.  cc  =  6.  23.  -  +  ^  =  c. 

3  a       b 

Solution. 

X  X 

-  -] —  =  c  (1)  =  given  equation. 

a       b 

bx  4-  ax  =  «5c  (2)  =  (1)  X  ah. 

(a-\-b)x=  abc  (3)         ( Vide  1'5,  ex.  3.) 

X  = 7  (4)  =  required  equation. 

a  -{-  b 

If  a  =  2,  b  =  3   and  c  =  10,  then  (ex.  14)  x  =  12. 

If  a  =  3,  6  =  4  and  c  =  14,  then  x  =  24. 

If  a  =  5,  6  =  6  and  c  ^  11,  then  x  =  30. 

If  a  =  7,  6  =  4  and  c  =  22,  then  x  =  56. 

mx        nx  .  abc 

24.  Solve  the  equation f-  -r-  =  c.  Ans.  x  = ; . 

a  b  bni  -\-  an 

If  w  =  3,  71  =  2,  a  =  4,  5  =  5  c^nd  c  =  23,  then  (ex.  16)  a:  =  20. 


82         EQUATIONS     OF     THE     FIEST     DEGREE. 

99.  To  solve  the  equation  ax  -\-  cl=  c  —  hx. 

By  axiom  (3)  we  may  subtract  d  from  both  sides  of  the  given 
equation  ;  thus  :       ax  -\-  d  —  d  =  c  —  hx  —  d.  (2) 

In  the  first  member  -f  d  now  cancels   —  d,  giving  the  equation 
ax  =  c  —  hx  —  d.  (3) 

By  axiom  (2)  we  may  now  add  hx  to  both  sides  of  equation  (3) ; 
thus ;  ax  -{-  hx  =  c  —  hx  -{-  hx  —  d.         (4) 

In  the  second   member  —  hx   cancels  -f-  hx,  giving   the   equation 
ax  -\-  hx  =  c  —  d.  (5) 

If  we  now  compare  (5)  with  the  given  equation,  we  see  that 

—  hx  has  been  transposed   to   the  first  member   of  (5),  its  sign 

being   changed    to    +  .     Also,    -f-  d,   of  the   given    equation,   has 

been    transposed    to    the  second    mem.ber  of   (5),   its   sign    being 

changed  to  —  .     The  value  of  x  is  now  found  by  96,  and  is 

c  —  d 
x  =  — —  .  (6) 

Hence, 

Tranq)Qse  the  terms  involving  x  to  the  first  member,  changing 
the  signs. 

Transpose  the  terms  not  involving  x  to  the  second  memher,  chang- 
ing the  signs.     After  this  proceed  as  in  96. 

EXAMPLES. 

1.  Solve  the  equation  Ox  —  5  =  40. 

Solution. 

9.x  —  5  =  40  (1)  =  given  equation. 

Transpose  -  5,  •)  9.^  =  40  +  5  (2) 

and  we  have  j  <  v.  y 

9x  =  45  (3)        by  96. 

.'^^  =  5  (4)  =  required  {([nation. 

2.  Solve  the  equation   10  +  3.t  ~  20  =  x  -]-  50. 


EQUATIONS     OF     THE     FIRST     DEGREE.         83 

Solution. 

10  4-  ?yx  —  20  =  a:  +  50  (1) 

3x  —  a;  =  50  —  10  +  20  (2) 

2x  =  60  (3) 

^  =  30  (4) 

3.  Solve  the  equation  15  +  re  —  20  =  5a;  —  T.t  +  40.  Ans.  x  —  15. 

4.  Solve  the  equation  I2x  —  50  +  GO  —  15.x  =  8x  —  70. 

Ans.       X    =     Tyy    . 

5.  Solve  the  equation  14a;  —  20  +  5  =  IG.r  -f  25  —  50. 

Ans.  X  =  5. 

(^  —  ^ 

6.  Solve  the  equation  ax  -\-  c  —  Ux  -\-  d.  Ans.  x  = . 

^  a  —  b 

If  a  =  20,   ?>  =  10,  c  =  60  and  d  =  80,  then  x  =  2. 
If  a  =  5,       6  =  4,      c  =  7    and  c/  =  8,     tlien  a;  =  1. 

26  —  5^ 

7.  Solve  the  equation  a;  +  oa  =  26  —  ca-.         Ans.  x=  — ■, 

1  -f-  c 

If  6  =  7,  a  =  1,  c  =  2,  then  x  =  3. 
If  6  =  9,  a  =  2,  c  =  3,  then  a;  =  2. 

IGO.  Hence,  to  solve  an  equation  of  the  first  degree  Avith  one 
u  nkn  0  \A'  n  qu  an  ti  ty  : 

(1.)    Clear  the  equations  of  fractions  h/j  multij^hjing  hj  the  leabt 

common  multiple  of  all  the  denominators.     (By  9§.) 
(2.)    Transpose   the   terms  involving   x   to   the  first  memhcr,   and 

those  not  involving  x  to  the  second  member.     (By  99.) 
(3.)    U7iite  the   terms  of  the  first   vicmher  so  as    to   indicate  a 
single  coefficient  of  x  (by  96),  and  reduce   the  terms  of  the 
second  memler  to  as  simp)le  a  form  as  possible. 

(4.)   Divide  the  equation  by  the  coefiicicnt  of  x. 

EXAMPLES. 

X      a;  4-  1       9 
1.  Solve  the  equation  -  —  ^ — - —  =  -  . 


84         EQUATIONS      OF     THE     FIEST     D  i:  G  li  E  E . 

Solution. 

2x  —  X  —  1  =  18  (1)    Vide  S9,   (5). 

;t==  19  (2) 

Vermcation    -— =  -^ . 

2  '±  ^ 

2.  Solve  the  equation  a; -j- a.-r  —  20  =  3a;  +  80.        Ans.  .t==33^. 

3.  Solve  the  equation  x  —  ^^- \- 20  =  4.r  —  29. 

A  o 

u4/«5.  ic  =  5. 
Solve  the  following  equations,  and  verify  the  result. 

X  X  X  ^  \  _  X  X  X  X 

4.  .  +  -  +  -+-  =  12-.  5.    .-------_-lT. 

111_1  J^  _1_  J_ 

^-    2"^3"^4""x*  •    10       x~  30* 

^^         3a;  —  1    ^   X  ,x+35 

8.  4a; \-  -  =  x -] {-  -  .  Ans.  x=^l. 

Jo  bo 

a;—  1       a;—  2       x       ^,  19 

9.  . ^ _  +  _  =  61  ---  .  .4,,..  .  =  Co. 


X  —  2       a;  —  4       ^       x  —  5 


10.    X  —  - — r; 1 p — = '<' H -; — •  Ans.x=0. 


x  —  7       a;  —  3       a;  —  4       ^  ^  9 

11.  a;  +  — ^ . p—  =  12-—- .  Ans.  x  =  17. 

5  4  0  10 

X  —  2       .T  —  4       ;/;  —  5       ^^3 

12.  X — ^ -— —  =  20-— .  Ans.  x  ==  GO. 

3  o  0  10 

x  —  2       a;  —  3       a;  —  4       ^11 

13.  X ::, z. —  =  6  — -  .  A71S.  X  =  20. 

0  4  5  20 

x  —  l?y       a;  — 40       ^,        3a;  —  2 

14.  X h  — ^—  =  21  +      .,.      .  ^«5.  X  =  28A. 

5  7  3d  !=> 

x-2       a;-4       a;  -  3       ^^29 

15.  X ^ — -  =  19  ^TTT  •  ^ns.  X  =  GO. 

o  o  o  oU 

a;— 2       a;  — 4       a;  —  7       a;  +  3         ^19 

7^8  2      ^     56  56 

3;7;  —  4       a;       a;       1 
!'•    — r^ — '^2'^'\:~2'  Ans.x=2, 


EQUATIONS     or     THE     Fill  ST     DEGREE.         85 

2  bx  4-  1 

18.  Solve  the  equation  7  —  x  —  -  (2x  -f  3)  =  6 . 

Solution. 

2                                5a;  +  1 
7  —  a?  —  -  (2x  -f  3)  =  6 —  (1)  =  given  equation. 

252  —  36a;  —  16a;  —  24  =  216  —  45x  —  9    (2)  =  (1)  x  36. 

Transpose  &  unite }        rj    9-1  ^o\ 

and  we  have  J   —  ^^  —  — -J-  C^j 

X  =  3  (4) 

19.  till  _|_  !^zd  +  4  To;  _  3^  =  68.  Am.  x  =  17. 

b  2  ' 

4-  Tx  4-  1^ 

20.  -  (x  +  2)  =  3  ^-^-.  .^>^5.  a;  =  7. 


21.    10  (.+  1)  -6.(^-1) 


x4/?5.  a;  =  2. 


4  /T.-r  — 9\         4  /a;+  17\        ,  ,  ^1 

22-    3.-5  H^j  =  5  (-3--)  +4-  ^-  ^  =  ^13- 


fit.'/;       134  ,         , ,       201k  —  268       6Ta;        ,  ,  13 

-^---(--1)  = 15 +  -1T-    ^«^.-  =  ley 


9  1  129  7 

24.  -(a:.-0-5(6-9^)  =  -j^.  Ans.x=.^.^. 

25.  |  +  5  =  |(..  +  2).  26.   |(^+1)  =  |(:^~1). 

101.  When  the  unknown  quantity  is  found  in  all  the  terms, 
involved  to  powers  no  two  of  which  differ  by  more  than  unity, 
the  equation  may  be  divided  by  x  involved  to  the  lowest  power, 
and  thus  reduced  to  an  equation  of  the  first  degree. 

EXAMPLES. 

1.  Solve  the  equation  —  —  x=.  — -. 

X  X 

Divide  by  x,  and  we  have  -t77  —  ^  =  "^  •       ■^^'^-  ^  —  ^^' 
■^      '  10  oO 


86  LITERAL     EQUATIONS. 

2.  Solve  the  equation  ——=  x^  —  5a;.  Ans.  a;  =  5— -. 

.       ^^2    ,       498.x'       332a;3  — 166^2 

3.  Solve  the  equation  27  -  x^  -\ -—-  = ~ . 

O  J.  D  0 

Ans.  0}  =  6. 

92       115  46  ,  ,^12 

4.  Solve  the  equation  _  +  —  = -^-^^ .   ^...  a:  =  10—. 

5.  Solve  the  equation  — -  =  135.^°.  Ans.  x  =  315. 

3?/  7?/2  1 

6.  Solve  the  equation  y  + r-  = —  A^is.  y  =  -. 

1  —  by       \  —  by  6 

.         —  4v  8y  ,  A 

7.  Solve  the  equation —  =  -— .  ^7i5.  y  =  4  - 

^  3  — 2y       15— 3/  o 


LITEEAL      EQUATIONS. 

102.  Literal  Equations  need  be  verified  only  by  introducing 
some  number  which  each  letter  may  be  made  to  represent  into 
the  given  equation,   together  with   the  corresponding  value  of  x, 

EXAJIPLES. 

6a^x       Sac       3h^x       4:bac       6hc       9h^x 
1.  Solve  the  equation  ^j-  +  -3 ^22^=  la"  +  Tl  +  "ll"- 

Solution, 

12a^x  +  33ac  —  Sh^x  =  45ac  +  125c  +  dh^x. 

12a^x  —  12h^x  =  12ac  +  12Z>c. 

(a2  _  52)  re  =  (a  +  h)  c. 
c 


a  —  h' 

If  a  =  5,  5  =  4  and  c  =  20,  then  x  =  20. 
Verification : 
6.52.20       3.5.20       3. 42. 20_  45.5.20       6.4.20       9.4^20 

~~n        ^        2  22~  ~        22         ^        n        ^  ~^2~~ 

If  a  =  7,  /^  =  5  and  c  =  30,  then  x  =  15. 


LITEEAL     EQUATIONS.  87 

cv  ^-~  ij       a  "—  -^  "^h 

2.  Solve  the  equation  — ^ -^ — \-  x=  b.     Ans.  x  =  — . 

If  a  =:  anythmg  and  Z>  =  8,  then  cc  =  6. 
If  Z>  =  12,  a:  =  9.      If  &  =  20,  a;  =  15,  &c. 

3.  Solve  the  equation ■\ —  -\ — -  =  -  . 

a  —  6       a^  —  6"*        a  -\-  b       x         „ 

a^  —  U' 


Ans.  X  = 


If  a  =  10  and  6  =  4,  then  cc  =  4.  *  2a  -j-  1 ' 

If  a  =  7     and  6  =  2,  then  a:  =  3. 

.  ah        X 

4.  Solve  the  equation  l\a  -\-  h)  x  =  a  -\-  — . 

9  6  (a  +  6)  +  4 

If  a  =:  2  and  6  =:  2,  then  x  =  -  . 

9 
If  a=  1  and  6  =  2,  then  x  =  — ^ . 

5.  Solve  the  equation  2x  -j \ 1 j —  =  a  -f  6  +  c. 

6a  +  86  +  9c 
Ans.  X  = — , 

If  a  =  11 J  6  =  11  and  c  =  11,  then  x  =  23. 

5 
If  a  =:  1,  6  =  2  and  c  =  3,  then  cc  =  4  —— . 

6.  Solve  the  equation  x —  =  a  —  -  . 

Z  o  o 

T/.  o        -,   7        r      ,  /^  ^"s.  a?  =  3  (a 6). 

If  a  =  8  and  6  =  5,  then  x  =  9.  ^  ^ 

If  a  =  3  and  6=2,  then  x  =  3. 

v.  Solve  the  equation 

x  —  a       X  —  6       x  —  c       41a  +  456  4-  46c 
^  +  — n-  +  — ^  + 


6  30 

-4ws.  a;  =  a  4-  6  +  c 
If  a  =  1,  6  =  2  and  c  =  3,  then  x  =  Q. 

8.  Solve  the  equation 

X  —  a^       X  —  a^c^       X  —  c"*  4a''       a^c^         c 

"^  +  ~7  5  35~  =   ~  35  +  T~  "^  17i' 

Ans.  X  = —^ —  . 

If  a  =  3  and  c  =  1,  then  x  =  2,  &c.  •^-' 


88  PKOBLEMS. 


r,     r.  ■,         ,                .        x  —  a^^x  —  ah  7a^         llah         bh^ 

9.  Solve  the  equation   — - — ■  -{ — ==  . 

X  —  0'' 

—  .  Ans.  x=z  (a  -{■  by. 

If  a  =  1,  Z>  =  2,  then  x=  9, 

1  U-  X               1  1 

1 0.  Solve  the  equation =  1  -f  -  ,  Ans.  x  = 


1  —  X  a  '  2a  -J-  1  * 

If  a=  1,  2,  3,  4,  &c.,  then  x=  ^,  |,  -] ,  J,  &c. 

11.  Solve  the  equation  = =  1  -}-  -  .  Ans.  x  =  — -^ — 

1  —  a  X  2a 

If  a=  ^,  l,  ^,  ij  &c.,  then  a;=  1,  2,  3,  4,  &c. 

103.  PEOBLEMS. 

(1.)  A  Problem  is  a  question  proposed  for  solution. 

(2.)  Any  algebraic  equation  can  be  considered  a  statement,  in 
algebraic  language,  of  the  conditions  of  some  problem. 

(3.)  Any  algebraic  problem,  if  properly  expressed,  can  be 
converted  into  one  or  more  algebraic  equations,  called  the  equa- 
tions of  the  proUem. 

(4.)  The  problem  is  solved  by  solving  the  equations.       ^1^  ^ 

(5.)  The  statement  of  a  problem  may  generally  be  effected 
by  considering  x  as  the  answer  sought,  and  indicating  the  oper- 
ations that  would  actually  he  perforfned,  if  the  value  of  x  were 
known,  in  the  verification  of  the  problem. 

EXAMPLES. 

1.  One -third  of  a  certain  number  is  7.     What  is  the  number? 
Let  X  =  the  number. 

Then  I  =  '^»  (!•) 

and  X  =   21.     {Vide  9':',  ex.  1.)     (2.) 

Now  if  we  perform  the  operation  on  21  which  is  indicated  on 

21 

X  in   equation   (1),  the  result  will  be  verified,  -—=7. 

o 


r  K  0  B  L  E  M  s .  89 

2.  If  Charles  had  twice  as  many  marbles  as  lie  now  has,  and 
also  three  times  as  many,  lie  would  have  as  many  as  John  and 
William  together,  the  former  of  whom  has  30,  and  the  latter 
half  as  many.     How  many  has  Charles? 

Let  X  =   his  marbles. 
Then,  2x  +  Zx  =   30  +  15. 

Whence,  x  =    9.  (T7t/e  96,  ex.  14.) 

Charles,  therefore,  has  9  marbles,  for  2  x  9  -f-  3  x  9  =  30  -f  15. 

3.  What  number  is  that  whose  half  and  third  added  together 
make  10?       {Vide  9S,  ex.  14.)  Ans.  12. 

4.  *  Three-fourths  of  a  number  added  to  two-fifths  of  it  make 
23.     What  is  the  number?       {Vide  9S,  ex.  16.)  Am.  20. 

5.  *If  a  number  is  added  to  its  half  and  third,  the  sum  will 
be  11.     What  is  the  number?      {Vide  9S,  17.)  Ans.  6. 

6.  '•■  If  the  fourth  of  a  number  be  subtracted  from  the  sum  of 
its  half  and  third,  the  result  will  be  7.  What  is  the  number? 
{Vid^c  9S,  18.)  An$.  12. 

7.  *If  one  forty-fifth  of  all  the  sheep  I  have  be  subtracted 
from  tlie  sum  of  two-fifths  and  four-ninths  of  them,  the  result 
will  be  37.  How  many  sheep  have  I?  (T7c/e9S,  19,  also  103, 
2.)  An^.  45. 

8.  If  from  nine   times  a  certain   number  5  be   subtracted,  the 

remainder  will  be  40.     AVhat  is  the  number?     (^Vide  99,  ex.  1.) 

Ans.  5. 

9.  ''If  the  fourth  of  a  certain  number  increased  by  1  is  sub- 
tracted from  half  of  the  same  number,  the  remainder  will  be  |. 
What  is  the  number?     {Vide  lOO,  ex.  1.) 

10.  *Four-t]nrds  of  a  number  increased  by  2  is  the  same  as 
three  halves  of  the  same  number  increased  by  1.  What  is  the 
number?     {Vide  iOO,  ex.  20.)  Ans.  7. 

o 


90  P  li  O  B  L  E  M  8  . 

11.  *If  5  be  added  (o  the  sixth  of  a  number  it  Avill  make  the 
same  thing  as  tlircc-fourths  of  the  number  increased  by  2.  What 
is  the  number?      (Vide  1®4>,  ex.  25.)  A/is.  6. 

12.  *If  from  a  number  its   half,  its   third,  and  three  more   be 

subtracted,  the  remainder  will  be  1.     What  is  the  number? 

Ans.  24. 

13.  *The  difference   between    the   fifth  and  sixth  of  a  number 

is  4.     What  is  the  number?     '^  —  '-  =  4.  Aiis.  120. 

5       6 

14.  flf  from  a  number  we   take  2,  and   divide  the  remainder 

X 2 

by  11,  the  quotient  will  be  6.     -— - —  =  6.  Ans.  68. 

15.  *  The  sum  of  two-thirds  and  three-fourths  of  a  number  is 
68.     AVhat  is  the  number?  Ans.  48. 

16.  ■]■  If  4  be  added  to  a  number,  one-third  of  the  sum  will 
be  5.     What  is  the  number.  Ans.  11. 

17.  flf  3  be  subtracted  from  a  number,  two-thirds  of  the 
remainder  will  be  16.     What  is  the  number?  Ans.  27. 

18.  *  In  one  flock  a  man  has  one-fourth  of  all  his  sheep,  in 
another  one-sixth,  in  another  one-eighth,  in  another  one-twelfth, 
and  in  another  450  sheep.  These  five  flocks  are  all  he  has. 
How  many  sheep  has  he,  and  how  many  in  each  flock  ? 

(1)  (2.)  (3.)  (4.)  (5.) 

Ans.  1200  =  300  -f  200  -f  150  +  100  +  450. 

19.  A  certain  nujuber  added  to  ten  times  itself  gives  132. 
What  is  the  number?  Ans.  12, 

20.  A  gold  watch  is  worth  ten  times  as  much  as  a  silver 
watch,  and  both  together  arc  worth  $132.  AVluit  is  each  watch 
worth?  Ans.  $120  and  $12. 

21.  A  man  paid  $74  for  a  sheep,  a  cow  and  an  ox.  The 
cow  was  valued  at  12  sheep,  and  the  ox  at  2  cows.  What  was 
the  price  of  each?  Ans.  $2,  $24,  $48. 


P  B  O  B  L  E  M  S  .  91 

22.  A  key  winds  both  a  gold  and  a  silver  watch.  The  silver 
watch  is  worth  twelve  times  the  key,  and  the  gold  watch  t\venty- 
five  times  the  key.  What  is  the  value  of  each,  if  all  are  worth 
$342?  Ans.  key  $9,  silver  watch  $108,  gold  watch  $225. 

23.  A  man  paid  $460  for  20  sheep,  5  cows  and  a  yoke  of 
oxen.  A  cow  was  valued  at  8  sheep,  and  an  ox  at  2  cows. 
What  was  the  price,  per  head,  of  each?     Ans.  $5,  $40  and  $80. 

24.  *  Three  men  and  two  boys  work  togetlier.  The  men  get 
a  quarter  of  a  dollar  per  day,  the  boys  one-fifth  of  a  dollar. 
How  many  days  must  they  work  to  receive  23  dollars? 

(^Vide  98,  ex.  16,  also  103,  (2)  and  ex.  4.)  Ans.  20  days. 

The  pupil  will  perceive  that  any  equation  may  be  that  of  an 
endless  variety  of  problems,  but  that  these  problems  are  only 
different  methods  of  expressing  the  sa^ne  conditions,  as  the  uniform 
statemeiit  proves. 

25.  A  starts  from  a  given  point,  and   travels  at   the  rate  of 

one  mile  per  hour.     After  an  absence  of  12  hours,  B  starts  after 

him  on  the  same  route,   at   the  rate  of  twelve  miles   per  hour. 

How  long  before  A  will  be  overtaken,  and  how  far  will  B  have 

traveled?  o  i  *- 

boLution. 

p 

Let  M  N  represent  the  road  traveled  over. 

Let  X  =   the  number  of  hours  required. 

Since  A  goes  one  mile  per  hour,  in  12  hours  he  will  go  12 
miles  =   M  P. 

Since  A  goes  one  mile  per  hour,  in  x  hours  he  will  go  x 
miles  =   P  N. 

Since  B  goes  12  miles  per  hour,  in  x  hours  he  will  go  \2x 
miles  =   M  N. 


92  TROBLEMS. 

Now  M  N  =  M  P  ■{-  P  N 

That  is         12x  =   12   +  x 

Whence  x  =   1  j'y   =   1  Lour  5  /j-  minutes. 

Now  B's  distance  being  12.t,  he  will  have  traveled  13  ^'y  miles. 

26.  The  hour  and  minute  hands  of  a  clock  are  together  at  12. 
"When  will  they  be  together  again  ?  Ans.  1   hour,  5  J^  min. 

27.  Two  men  start  from  the  same  point,  and  travel  in  the 
same  direction  ;  the  first  steps  twice  as  far  as  the  second,  but  the 
second  makes  five  steps  while  the  first  makes  one.  At  the  end 
of  a  certain  time  they  are  300  feet  apart.  How  far  has  eacl 
traveled?      2|x=  300  +  x.  Ans.  V  200,  2'"^  500  feet- 

28.  Two  men  start  from  the  same  point,  and  travel  in  ojiposite 
directions ;  the  first  steps,  each  time,  two-thirds  the  distance  of 
the  second  ;  but  the  second  makes  only  4  steps  while  the  first 
makes  7.  At  the  end  of  a  certain  time  they  are  520  feet  apart. 
How  far  has  each  traveled?  A?is.  V*  280,  2*^^  240. 

29.  A  cistern  has  three  pipes.  The  first  will  fill  it  in  2  (li) 
hours,  the  second  in  3  (3|)  hours,  the  third  in  4  (5)  hours. 
In  what  time  will  the  cistern  be  filled  when  the  three  pipes  are 
running   together.  ^^^^^^.^^^^ 

Let  X  =   the  time ;  then, 
I  r=   the  part  the  1st  will  fill  in  one  hour. 
1  =         "        "        2nd      "        "        "        " 
1  =        "        "        3rd      '*        "        "        " 

4 

L  =   the  part  all  will  fill  in  one  hour. 

X 

Hence,  (axiom  1)  |  +  |  +  i=-j  whence  x  =:  i|  of  an  hour. 

(Vide  lOO,  ex.  6.) 

30.  Solve  the  above  problem  using  the  numbers  in  the   (     ) . 

Ans.  48  minutes. 

31.  A  cistern  has  three  pipes,  two  at  the  top  and  one  at  the 
bottom.     One  of  the  top  pipes  would  fdl  it  in  5  hours,  the  other 


PROBLEMS.  93 

in  6;  but  the  pipe  at  the  bottom  empties  it  in  8|  hours.  In 
what  time  will  the  cistern  be  filled  when  the  pipes  are  running 
together'?  Ans.  4  hours. 

32.  A  can  do  a  piece  of  work  in  3  (2|)  days,  B  in  5  (2) 
days,  and  C  in  7|  (8)  days.  In  what  time  can  they, all  do  it 
by  working  together?  A?is.  lA  days. 

33.  A  and  B  can  do  a  piece  of  work  in  5  (f^)  days.  A  can 
do  it   alone  in  7    (Ti)   dciys.     In  what   time   can  B  do   it   alone? 

Ans.  17^  days. 

34.  A  and  B  can  do  a  piece  of  work  in  5  days ;  A  and  C 
in  6a  days;  B  and  C  in  7  days.  In  what  time  would  all  do  it 
by  working  together?      ^  -|-  ^  -j-  j^  =  i.  Ans.  4  days. 

35.  A  man  and  his  wife  could  drink  a  cask  of  beer  in  10 
days.  In  the  absence  of  the  man  it  lasted  his  wife  30  days. 
How  long  would  the  man   be  occupied  in    drinking  it? 

Ans.  15  days. 

36.  A,  B  and  C  could  do  a  piece  of  work  in  A  days;  A,  B 
and  D  in  I  days;  A,  C  and  D  in  i|  days;  B,  C  and  D  in  i| 
days.  In  what  time  could  they  all  do  the  work,  and  in  what 
time  could  each  man  do  it  alone?  Ans.  All  in  |?  days; 

A  in  1  ;    B  in  2  ;    C  in  3  ;    and  D  in  4  days. 

37.  Divide  55  (80)  into  two  parts,  so  that  the  less  (greater) 
part  divided  by  the  difference  (sum)  of  the  parts  shall  be  2   (|). 

Ans.  33  and  22. 

38.  Four  places  are  situated  in  the  order  of  the  four  letters 
A,  B,  C  and  D.  The  distance  from  A  to  D  is  134  miles.  The 
distance  from  C  to  D  is  |  the  distance  from  A  to  B,  and  i  the 
distance  from  A  to  B  added  to  half  the  distance  from  C  to  D  is 
three  times  the  distance  from  B  to  C.     What  are  the  distances? 

39.  A  person  went  to  a  tavern,  where  he  spent  5  shillings, 
and  then  borrowed  twice  as  much  as  he  had  left.     He  does  the 


94  P  11  O  B  L  E  M  S  . 

same  at  a  second  and  a  third  tavern ;  but  on  spending  21  shil- 
lings at  a  fourth  tavern  he  had  nothing  left.  How  much  had 
he  at  first?  Arts.  8  shillings. 

40.  A  boy  had  a  number  of  marbles.  He  laid  aside  2,  and 
then  w^oi;;i  in  play  as  many  as  he  had  left.  He  then  laid  aside 
3  more,  and  again  won  as  many  as  were  left.  He  now  adds  4 
more  to  the  reserved  pile,  and  wins,  as  before,  as  many  as  he 
has  left.  Then  counting  his  marbles  he  finds  13.  How  many 
did  he  begin  with?  Ans.  5. 

41.  Two  boys,  Charles  and  John,  play  marbles.  First  game, 
Charles  wins  4  marbles.  Second  game,  John  wins  12.  Charles 
again  wins  4  in  the  third  game,  and  John  wins  6  in  the  fourth 
and  last  game.  John  now  has  three  times  as  many  marbles  as 
Charles,  although  each  had  the  same  number  Avhen  the  play 
commenced.     How  many  marbles  had  each  at  first  ?        Ans.  20. 

42.  A  commenced  trade,  and  at  the  end  of  the  third  year  found 
his  original  stock  tripled.  Had  his  gains  been  $1000  per  year 
more  than  they  actually  were,  he  would  have  doubled  his  stock 
each  year.     What  was  his  original  stock?  Ans.  $1400. 

43.  Divide  the  number  20  into  two  parts,  so  that  the  product 
of  the  parts  shall  be  5  times  the  greater  part. 

Let  X  =   the  greater  part,  and  20  —  x  the  less. 

Then  20.^;  —  x^  =  5.t,  whence  15  and  5  are  the  numbers. 

44.  Divide  the  number  40  into  two  parts,  so  that  the  product 
of  the  parts  may  be  35  times  the  smaller  part. 

45.  A  boatman  rows  14  miles  an  hour  with  the  tide.  Against 
a  tide  two-thirds  as  strong  he  rows  only  4  miles  an  hour.  What 
is  the  velocity  of  the  tide  in  each  case?        Ans.  G  and  4  miles. 

46.  Three  persons,  A,  B  and  C,  were  seen  traveling  in  the 
same  direction.     At  first  A  and  B  were  together,  and  C  12  miles 


PROBLEMS.  95 

in  advance  of  them.  A  goes  7,  B  10,  and  C  5  miles  per  hour. 
In  what  time  Avill  B  be  half  vrnj  between  A  and  C?  How  long 
before  C  will  be  midway  between  A  and  B?  Plow  long  since 
A  was  midway  between  B  and  C? 

Ans.  respectively  Ih.  30m.,  oh.  2obn.,  and  12/i. 

104.  (1.)  It  is  often  much  more  convenient  to  represent  the 
unknown  quantity  by  such  an  expression  as  will  avoid  the  intro- 
duction of  fractions  into  the  equation  of  the  problem.  It  is  in 
fact  a  much  better  exercise  to  solve  a  single  problem  in  several 
different  ways  than  to  be  engaged  on  as  many  different  problems. 
The  shortest  method  of  solution  should  always  be  found  out,  as  it 
leads  to  the  clearest  insight  into  the  problem. 

EXAMPLES. 

1.  What  number  is  that  whose  half  and  third  added  together 
make  10?     (Vide  103,  ex.   3.) 

Let  6x  =   the  number. 
Then  3x  +  2x  =10,  whence  x  =  2  and  6x  =  12. 

In  the  same  way  solve  those  marked  *  in  the  preceding  sec- 
tion ;  {.  e.,  let  the  unknown  quantity  be  represented  by  the  least 
common  multiple  of  the  denominators  of  the  fractions  in  the 
problems. 

2.  The  rent  of  an  estate  is  this  year  5  per  cent,  {.  e.  ^q,  greater 
than  it  was  last  year.  This  year  it  is  8-10  dollars.  What  was 
it  last  year'?     Let  20a;  =  the  rent  last  year.  Ans.  $800. 

3.  If  from  a  number  we  take  2,  and  divide  the  remainder  by 
11,  the  quotient  will  be  6.     What  is  the  number? 

Let  llx  -f  2  =  the  number. 
Then  x  =  6,  and  llx  -f  2  =  68.     (Vide  103,  ex.  1-1.) 

In  a  similar  manner  solve  those  marked  f  of  section  103. 


96  PROBLEMS. 

4.  Wliat  number  is  that  from  which  if  5  be  subtracted  |  of 
the  remainder  will  be  40?     3x -f  5.  Aiis,  65. 

5.  What  number  is  that  to  which  if  7  be  added  |  of  the  sum 
will  be  18?     3x  —  7.  A71S.  20. 

6.  A  teacher  spent  |  of  his  salary  for  board  and  lodging,  1  of 
the  remainder  for  clothes,  i  of  what  remained  for  books,  and 
saved    $120    per  annum.     AVhat   was   his   salary  ?      15.7?  =  salary. 

Ans.  ^360. 

7.  In  a  mixture  of  wine  i  the  whole,  plus  25  gallons,  was 
wine;  1  the  whole,  minus  5  gallons,  was  water.  What  was  the 
quantity  of  each  in  the  inixture? 

Let  6x  =  the  whole,  then  ox  +  25  -f  2.^  —  5  =  6x. 

8.  One-half  of  a  certain  number  is  the  same  as  |  another 
number.  But  if  5  is  added  to  the  first  and  10  to  the  second, 
then  I  of  the  first  is  the  same  as  |  of  the  second.  What  are 
the  numbers'?     2x  and  3x.  Ans.  20  and  30. 

9.  Divide  00  into  four  parts  so  that  if  the  first  be  diminished 
by  2,  the  second  increased  by  2,  the  third  divided  by  2,  and  the 
fourth  multiplied  by  2,   the  results  will  be  equal. 

Let  2x  =  tlie  quantity  to  which   they  are  to   be  equal. 

1st  part.  2(1  part.  2d  part,        ith  part. 

Then  2x  +  2  -f  2x  —  2  +    4x    -}-      x     =  90,  whence  x=  10. 
And         22  18  40  10    are  the  parts. 

10.  Divide  the  number  151  into  5  parts  so  that  twice  the  1st, 
one-half  the  2d,  one-third  the  3d,  one-fifth  the  4^/?,  and  three 
and  one-half  times  the  5th  shall  be  equal. 

Let  14:X  =  the  quantity  to  which  they  are  to  be  equal. 

11.  A  person  supported  -himself  3  years  for  $50  a-year.  At 
the  end  of  each  year  he  added  to  that  part  of  liis  stock  which 
was  not  thus  expended  a  sum   equal   to  I  of  this  part.     At  the 


PKOBLE^IS. 


97 


end  of  the  third  year  his  original  stock  was  doubled.     What  was 
the  amount  of  stock  at  first? 

Let  27x  +  200  =  the  original  stock. 

Then  27x  +  150  =  the  original  stock  less  $50. 

And  Qx  -\-    50  =  one-tliird  this  remainder. 

36a:  +  200  =  stock  at  the  close  of  first  year. 
In  the  same  way  48x  +  200  =  "         "         "        second     " 

And  64a;  +  200  =  "         "         "         third      " 

Therefore    64a;  +  200  =  54a;  +  400  by  the  question. 

Whence  a;  =  20 

And  27a;  +  200  =  $740  the  original  stock. 

12.  From  a  certain  sum  of  money  I  took  one-third  part,  and 
put  in  its  place  $50.  From  this  sum  I  took  one-tenth  part, 
and  soon  replaced  it  with  $37,  when  the  sum  amounted  to  $100. 
What  was  there  at  first?     15a;  —  75. 

13.  A  tree  80  feet  high  was  broken  by  the  wind  in  such  a 
manner  that  the  top  reached  the  ground  just  40  feet  from  the 
bottom  of  the  tree.     How  high  up  was  the  tree  broken? 


A  40 

Let  X  =  the  distance  from   the  bottom,  and  80  —  a;   the  part 

broken  off. 
Then  (80  —  a;)^  ==  a;^  -f  40=  Euclid,  Bk.  I,  47. 

Or      6400  —  160a;  +  x^  =  x^  +  1600 

a;  =  30  =  A  B.     80  —  a;  =  50  =  B  C. 


98        EQUATIONS     OF     THE     F 1 11 B  T     DEGREE. 

14.  Two  trees  80  and  60  feet  high  stand  on  the  tame  hor- 
izontal plane,  100  feet  apart.  Where  must  a  person  stand  to  be 
equally  distant  from  the  top  of  each?      (^Vide  121,  ex.  4.) 

Ans.  G-4  feet  from  the  shorter  tree,  or  36  feet  from  the  taller. 

EQUATIONS   OF   THE   FIRST    DEGREE    INVOLVING   TWO 
UNKNOWN   QUANTITIES. 

105.  Simultaneous  equations  are  those  in  which  the  values  of 
the  unknown  quantities  are  the  same  in  both.     Thus, 

X  -j-  7/  =  ^0  and  x  —  j/  =  6 
are  simultaneous  equations,  because  either  of  them  can  be  verified 
when  a;  =  18  and  ?/  =  12. 

106.  Simultaneous  equations  are  independent  of  each  other 
when  one  is  not  a  mere  transformation  of  the  other,  or  when 
one  equation  is  not  a  result  of  the  combination  of  two  or  more 
equations. 

Thus,  a;  -f  y  =  30  and  x  —  j/  =  6  are  independent  simultane- 
ous equations ;  but,  x  -^  y  =  o^  and  3a;  =  90  —  ?>y  are  depend- 
ent, since  the  first  may  be  easily  obtained  from  the  second. 

Also,  3x  +  2j/  +  ;i'  =  10,  X  +  ?/  +  z  =  6  and  x-]-2y-\-Zz=  14 
are  dependent,  since  the  second  is  one-fourth  the  sum  of  the 
other  two. 

ELIMINATION. 

lOf .  Elimination  is  the  operation  of  combining  two  equations 
in  such  a  manner  as  to  cause  one  of  the  unknown  quantities  to 
disappear  in  a  new  equation. 

There  are  three  principal  methods  of  elimination,  by  addition 
or  subtraction,  by  suhstitution,  and  by  comiiarison. 


ELIMINATION.  99 

ELIMINATION   BY   ADDITION   OR   SUBTRACTION. 

1©§.  1.  Eesume  the  equations, 

0^  +  7/  =  30  (1) 

and  X  —  3/  =  C  (2) 

By  axiom  2,  we  may  add  these  equations  together.     Doing  so 
we  have  2x  =  36       (3)        =  (1)  +  (2).      Vide  34,  ex.  36. 

Whence  a;  ==  18       (4)       =  (3)  -v-  2. 

By   axiom   3,  we  may   subtract   (2)    from    (1).      Doing  so   we 
have  2y  =  24       (5)       =  (1)  —  (2).      Vide  45. 

Whence  y  =  12       (6)       =  (5)  -^  2. 

By  putting  the  values   of  x  and  7/  in    place   of  these   letters 
in  (1)  and  (2),  we  have  18  -f  12  =  30 
and  18  —  12  =  6  Vide  105. 

In  the  same  way  find  the  values  of  x  and  y  in  the  followino* 
sets  of  equations. 
iB  +  ?/=10  X  -i-  y  =12 

X  —  3/  =  4  X  —  y=8 

2.  Again  ;   take  the  equations, 

3x  +  2y  =  22 

and  2x  +  3y  =  23 

By  axiom  4,  we  may  multiply   (1)   by  2   and   (2)   by  3. 
This  gives  Gx -}-  4y  =z  44  (3)  =  (1)  x  2 

and  6x+  9y=  69  (4)  =  (2)  x  3 

By  axiom  3,  subtract   (3)  from  (4),  and  we  have, 

%  =  25  (5)  =  (4)  -  (3) 

Whence  y  =  5  (6)  ==  (5)  -v-  5 

By  axiom  4,  we  may  multiply   (1)  by  3  and   (2)  by  2. 
This  gives  9x  +  6y  =  66  (7)  =  (1)  x  3 

and  4x  -f  6y  =  46  (8)  =  (2^)  X  2 


x-\-  y  =  20 

a;  +  y  =  25} 

X  — y  =  15 

x—^  =  3l 

(1) 

(2) 

100  ELIMINATION. 

By   axiom  3,   subtract   (8)   from   (7),   and  we  have, 

5x  =  20  (9)  =  (7)  -  (8) 

Whence  a;  =  4  (10)  =  (9)  -h  5 

By   putting   the   vakies   of  x  and   y   in   phice   of  these    letters 
in   (1)  and   (2),  we  have  3.4  +  2.5  =  22 
and  2.4  +  3.5  =  23 

wliich  proves  that  the  values  of  x  and  y  are  correct. 

We  multiplied  equation  (1)  by  2,  and  equation  (2)  by  3, 
simply  to  make  the  coefficients  of  x  in  these  equations  alike,  and 
because  the  signs  before  the  like  coefficients  of  equations  (3)  and 
(4)  are  alike;  by  subtracting,  x  disappears  in  the  resulting  equa- 
tion (5),  where  there  is  only  the  letter  y,  whose  value  in  (6) 
is  obtained  in   the  manner  heretofore   explained. 

We  now  multiply  equation  (1)  by  3,  and  equation  (2)  by  2, 
to  make  the  coefficients  of  y  alike,  which  leads  to  the  value  of  x, 
in  the  very  same  way  as  before. 

By  this  process  we  have  eliminated  x  and  found  the  value  of  y. 
We  then  eliminated  y  and  found  the  value  of  x. 


3.  Again ;  take 

the  equations, 

5^  +  3j/  =  13 

(1) 

and 

3.T  —  7?/  =  —  1 

(2) 

By  ax.  4, 

35x  +  2ly  =  91 

(3) 

=  (1)  X  7 

By  ax.  4, 

9x  —  21y  =  —  3 

(-i) 

=  (2)  X  3 

By  ax.  2, 

44.x  =  88 

(5) 

=  (3)  +  (4) 

By  ax.  5, 

x=2 

(C) 

=  (5)  -  44 

By  ax.  4, 

Vox  +  9y  =  39 

0) 

=  (1)  X  3 

By  ax.  4, 

Ihx  -  35?/  =  —  5 

(8) 

=  (2)  X  5 

By  ax.  3, 

44y  =  44 

(9) 

=  (7)  -  (8) 

By  ax.  5, 

7/=l 

(10) 

=  (9)  _4_  44 

In  this  example,  after  making  tlie  coifficienis  of  y  alike,  because 
the   sigiis   of  these   coefficients   are   unlike,    we   add  equations   (3) 


ELIMINATION.  101 

and   (4),  and   y  disappears.     In   other  particulars   the  steps  arc 
the  same  as  in  the  previous  example. 

4.  Take  the  er[uations,  '    \  \  \  \  ]   \  ^ 

2  +  3-"^       ^^^ 

and  ^+|  =  3|       (2) 

By  ax.  4,      3x  +  2^  =  22      (3)       =  (1)  x  G    (  VicU  9S.) 
By  ax.  4,      2x  +  3^^  =  23       (4)       =  (2)  x  6     {Vide  9S.) 

Equations  (3)   and  (4)  are  the  same  as  (1)   and   (2)  of  ex.  2. 
They  should  be  treated  in  like  manner. 

5.  Take  the  equations, 


X        y        ^          y 

-    4-    —     =    1-3-   —    - 

2  ^  10         ''       5 

0) 

and 

X      y            1  +  X 
7       2               14 

(2) 

By  ax.  4, 

5.x-  +  ?/  =  13  —  2y 

(3) 

=  (1)  X  10 

By  ax.  4, 

2x  —  7^  =  —  1  —  re 

W 

=  (2)  X  14 

Vide  99, 

hx-\-oy=  13 

(5) 

=  (3)  transposed. 

Vide  99, 

3x  -  7y  =  -  1 

(6) 

=  (4)   transposed. 

The  equations    (5)    and   (6)    are   (1)   aiid   (2)   of  ex.   3,   and         | 
should  be  treated  in  the  same  manner.     Therefore,  j 

109.  Having  two  equations  with  two  unknown  quantities,  to 
find  their  values  by  the  method  of  elimination,  by  addition  or 
subtraction :  I 

(1.)  If  necessary,  clear  the  equations  of  fractions. 

(2.)   In   each   equation   collect   all   the   terms  involving  x  into 
one  term,  and  write  this  term  first  in  the  first  member  of  a  new         ! 
equation,  prefixing  the  correct  sign.  i 

(3.)  In  each  equation  unite  all   the  terms  involving  y  in  one 


102 


E  L  I  ^I I  N  A  T  I  0  N  . 


terra,  and  write  this  term  second  in  the  first  member  of  the  new 
equation,  prefixing  the  correct  sign. 

> .  ('4.)  Ii'  each  equation  collect  the  known  quantities  into  one 
term,  and  write  this  term  in  the  second  memher  of  the  corres- 
.pOTiding  ,iic<v*  eqviQtipn,  prefixing  the  correct  sign. 

(5.)  Find  the  least  common  multiple  of  the  coefficients  of  the 
letter  you  wish  to  eliminate,  and  multiply  the  equations  respect- 
ively by  the  quotient  of  this  multiple  divided  by  the  coejjlcient 
of  the  same  letter  in  the  equation  to  be  multiplied.  The  result- 
ing coefficients  of  this  letter  will  be  the  same. 

(6.)  If  then  the  signs  of  these  coefficients  are  alike,  suhtract  one 
equation  from  the  other ;  if  the  signs  of  these  coefficients  are  unlike, 
add  one  equation  to  the  other.      (^Vide  40  and  34.) 

(7.)  From  the  resulting  equation  find  one  of  the  unknown 
quantities,  and  by  repeating  the  steps  5   and  6,  find  the  other. 

These  directions  should  be  followed  till  the  process  is  perfectly 
at  the  command  of  the  pupil.  The  notation  on  tlie  right  should 
be  invariably   demanded,  since  it  leads  to  habits  of  accuracy. 


EXAMPLES. 


1.  Find  X  and  y  in  the  equations 

x-\-  8 


and 


21-6y  = 


23  —  5.T 


4 
3 


(1) 


(2) 


8i-2ii/  =  x  +  S 

(3) 

69  —  15x  =  3/  +  6 

(4) 

x  +  24ij  =  70 

(5) 

lox  +  y  =  fi3 

(6) 

15x  +  360)/  =  1140 

(7) 

35%  =  1077 

(8) 

y  =  3 

(9) 

=  (1)  X  4 

=  (2)  X  3 

=  (3)  transposed 

=  (4)  transposed 

=  (5)  X  15 

=  (7)  -  (6) 
=  (8)  -^  359 


E  L  I  ]M  I  N  A  T  I  0  N  . 


103 


360a)  +  24y  =  1512 
359x=1436 
rc  =  4 


2x  +  ry=     38  > 
3x  —  5^  =  —  5  ) 

nx^Zy=    10^ 
'^'    8x  +  2y=120  5 

5x  -  4j/  =    7  > 
10x4-5^  =  40  5 

7x-    %=-2^ 
^'   8x  +  21j/=     29  > 

12a:  -  7?/  =  12  ^ 
llx-33/=ll5 

^     4x  +  3j/  =  65> 
^-    5a:_23/=41  5 

g     3a) +  53/=  15  > 
4a;  4-    y=    3  ) 

7x+    5^  =  2^ 
-^^    i4^__10j/=0  5 


6. 


10. 


11. 


12. 


3x      y  ^c, 

52  "^   44       " 


(10) 

(11) 
(12) 


(6)  X  24 
(10)  -  (5) 
(11) -f- 359 


13. 


•"^  _  ^  =  _  90   ^ 

7       10 


1+   3^=     131  J 


14. 


^  =  1  +  20 
2       4^ 


^ 


5    ^3       4^1 


15. 


y_^ 

7       6 


1 


^  +  ^=1^1 


16. 


2  ! 


5.'.-|  =  29      ] 


17. 


X  — 


.y-1 


J 


x  +  y  ,^-y  r^^^l] 
■~2~  "^  ""2~  "^  3     4  I 

3"^4  5    "         7      ^ 


19. 


2"^3 
3^4 


1 


104 


ELI  j\I  I  NATION     BY     SUBSTITUTION. 


20. 


-  +  -=.-: 
X       y       b 


21, 


2x        Zy 


26 


3x 

4.7; 


^  ~        J 


22. 


+  ^=20 

7    ^  14 


14         7    -       ^^  J 
1 


* 


f  +  T^  =  99   I 


7x  +  |=51 


^ 


24. 


5.7;        3?/ 


25. 


y_ 

41) 

^  =99  f 


49 


51 
99 


*  15 

26. 


17 


27. 


*    5 


^  7 


09 


2.x 
T+   5 


1631     I 
35     J 


28: 


4x 

?>x 


By 

T 

Ay 
5 


2^*4x  +  3j/  =  3> 
3;c  -}-  4^  ==  4  > 


30. 


4:X        3y 

3.7;        4y 
"8~"^  T 


37 
42  J 


*  40^  +  3^  =  7-^ 
3a;  +  4y  =  7  5 


ELIMINATION     BY     SUBSTITUTION. 


110.  1.  Resume  tlie  equations, 

x  +  y  =  SO  (1) 

and                           x—y=6  (2) 

Transpose  y  in  equation   (2),  and  we  have 

x=6-]-y  (3) 


ELIMINATION     BY     SUBSTITUTION.  105 

Substitute  this  value  of  x  for  x  in   equation   (1),  and  we   have 
G+y  +  y  =  30  (4) 

whence  ?/=-12  (5) 

Substitute  this   value   of  y  for  y  in   equation   (1),  and  we   have 
.T  +  12  =  30  (6) 

♦vlience  re  =  18  (7) 

2.  Take  the  Cvquations,     "^  +  |  =  6|  (1) 

and  I  + 1  =  ^^  ^^^ 

Clear  (1)  of  fractions     5x  +  3j/  =  95         (3)       =  (1)  x  15 
Clear  (2)  of  fractions     3x  +  2y  =  60         (4)        =  (2)  x  6 

60  —  23/ 
Find  a:  in  (4)  x  = ^  (o) 

Substitute  this  for  x  in  (3) -|-  3y  =  95  (6) 

o 

whence  3/  =  1^  C'^) 

and     cc  =  ^^^^^  =  10     (8)     =r  (5)  in  which   2^/=:  30 

It  is  evident  that  these  steps  may  be  taken  on  any  two 
equations,  hence, 

111.  Having  two  equations  with  two  unknown  quantities,  to 
find  their  values  by  tlie  method  of  elimination  by  substitution, 

(1.)  If  necessary,  clear  the  equations  of  fractions, 

(2.)  Find,  in  either  of  the  equations,  the  value  of  one  of  the 
unknown  quantities  in  terms  of  the  other,  and  substitute  this  value 
for  the  same  unknown  quantify  in  the  other  equation. 

(3.)  From  the  equation  thus  formed,  find  the  value  of  the 
letter  involved. 

(4.)  Substitute  this  last  value  for  the  letter  to  which  it  is 
equal  in  any  equation  excejjt  that  from  which  it  was  obtained,  and 
find  the  value  of  the  otlicr  letter. 


106 


ELIMINATION     BY     SUBSTITUTION. 


EXAMPLES. 

3.  Find  x  and  y  in  the  equations, 


=  4 


5x        X  —  y 

IT         T~ 

and  2x  -f  3j/  =  43 


20a3  — llx+  lly  =  176 

^x  +  llj/  =  176 
43  — 3y 


X  = 


387  —  27y 
■        2 
whence 


+  lly=17G 


and 


43  —  21 


2 


=  11 


(1) 

(2) 
(3) 
(4) 

(5) 

(6) 
(7) 

(8) 


=  (1)  X  44 
=  (3)  reduced 

=  (2)  v{(h  above,  2 

^  (4)  vide  above,  2 
vide  above,  3 
vxde  above,  4 


3a.  +  4y=18^ 
^-    2x-    y=    ly 


2x  —  3?/ 


rr    92 


9. 


5* 


4x 


+  3j/=16^ 
3x  +  4^  =  19  5 


y  — 1     . 


y     ^ 


6. 


10. 


7. 


8. 


4      51 


»  +  |-10  =  | 


i-l^^y-m 


V       ^       ^ 
5       5^ 


^       V  ^ 

-  +^i  =  2 
5^4 

-  —  ^  =  —  1 


a;_2       10  — a;_y  — 10    ] 

11."^  3---^    ^ 

2^  +  4  _  2:r  +  y  ^  0^  +  13  j 
3  8  4       J 


12. 


3x  +  4^  _  40  —  X 1 
5  4        ' 

o,^       2.y^84-7/| 
"■■''        o  6       J 


^      ELIMINATION      BY     SUBSTITUTION. 

^  ,  y 


107 


13. 


a^     y     o 

O  -J 


a?  +  y  ■  ^— y^  q1 


r 


14. 


10 


2 


•^  +  y  ,  ^  —  y  ^  -^ 


^ 


2 


2.x  —  ?/       3       oy  —  AiX 
-  -  - 


15. 


x-^y 


9  2 

•^  3 


16.  Find  a;  and  y  in  the  equations, 

(1) 


9         9 

-  +  -=10 
a;      y 


'i 


1  _3_ 
X      y 

X  -\-  y  =  ^xy 
y  —  ox=  —  7xy 
x  —  bxy  =  —  y 
(1  ^by)x=  —y 

y 


X  = 


y  + 


3y 


l-5y 

,2 


(2) 

(3) 
W 
(5) 
(«) 

(V) 


=  (1)  X  a-y  and    -^  by  2 

=  (2)  X  xy 

=  (3)    transposed 

=  (5)   factored 

=  (6)  27V/c^  above,  2 


ry 


1-57/       1-5^ 
whence  y  =  ^,   x  =  ^ 
1 


(8)  =  (-15  r?V/e  above,  2 


4       3^. 

-+-  =  3  I 

^^•^  S       5 


^ 


X  =  1  |i 


19.t 


5 


y  =  -l 


-+  - 

a;  y 
5_6 
.X        ?/ 


rzV/e  1®1,  ex.  6. 


87 


18.t 


-=i    ]x=2H 
x      y 

6       10       ,  , 
X       y  ' 


20.t 

'  6      10       ,  I 

-+    -=:4 

^       y  J 


108  ELIMINATION. 

ELIMIXATIOTT     BY     COMPARISON. 

112.  1.  Find  the  values  of  x  and  y  in  the  equations, 

a;  +  y=30  (1)  and  x—y  =  6  (2) 

Transpose  y  in  each  of  these  equations,  and  we  have 

x=30  —  ?/         (3)         and  x  =  6 -j- y  (4) 

Now  by  ax.  1   tiicse  values  of  x  must  be  equal,  that  is. 

By  comparison  30  —  y  =1  Q  -\-  y  (5) 

whence  y  =  12  (C) 

and  from   (3)   or   (4)         .t=18  (7) 

2.  Find  the  values  of  x  and  y  in  the  equations, 

l  +  ?  =  2  (1)        and         ?-l^=-l         (2) 

Ay  +  3.C      =  2.ry         (3)  8y    _  15.^;  ==  —  .tj/     (4) 

3x  —  2.TJ/     =  —  4y     (5)  lox  —  .^y     =  8y  (G) 

(3  -  2y)  X  =  -  4y     (7)  (15  -  y)  rr  =  8j/  (8) 

3  — 2j/  ^  ^  15—3/ 

—  4y  8y 

By  comparison V- =  t^- (H)      ^"^^^^  ICl,  ex.  7. 

•^         ^  ^V-23/       lo-y  ^ 

whence  ?/  =  4i  and  x  =  3i. 

It  is  evident  that  these  steps  mny  be  taken  on  any  two  equa- 
tions, hence, 

113.  Having  two  equations  with  two  unknown  quantities,  to 
find  their  values  by  the  method  of  elimination  hy  comjmrison : 

(1.)  If  necessary,  clear  the   equations  of  fractions. 

(2.)  Find,  in  both  equations,  the  value  of  the  same  unknown 
quantity,  in  terms  of  the  other,  and  make  these  values  equal. 

(3.)  From  the  equation  thus  formed,  find  the  value  of  the  let- 
ter involved. 

(4.)  Same  as  110,  4. 


ELIMINATION     BY     CO  :\I  PARI  SON. 


109 


3. 


4. 


4x  +  3y  =  7  ^ 
X  -^  y  =  2'^ 

^  +  2o  =  x 

x  +  y 


—  o 


y 


g     x  +  2  =  ^y} 
•   3/  +  4  =  |x     3 

Src-f  2  =  14?/ 
9.    a;  +y 

2 


91 

—      4" 


—  i-i^X 


a;  +  25  =  ^y 


10. 


0. 


X 


7/  +  25  =  -  +  15 


10a;  +  y  =  4  (x  +  3/)  > 

10a; +  3^  +  18  =  \^y  -\-xS 


6. 


:.  +  |^  =  iml 


ii.f 


.;c        3y 
4"^  20 


^ 


27^ 


?+^  =  2 
X      y 

- 

4        5 

a;        3y        ^J 

'    +'    =2      ^ 
3a;        43/ 

1          1 
2a;       y 

■^ 

12.t 


114.  Either  of  the  three  methods  of  elimination  may  be  em- 
ployed to  solve  equations  consisting  of  tAvo  unknown  quantities. 
Practice,  however,  and  repeated  efforts  to  do. so,  will  enable  the 
student  greatly  to  abridge  the  work  in  almost  every  case  that 
can  happen.  To  illustrate  this  remark,  we  will  repeat,  by  all 
the  metliods,  the  solution  of  ex.  16    111. 


.  The  equa- 
tions are 

2       2 

-+-=10 
X      y 

(1) 

and 

i_?=-7 
X      y 

(2) 

1       1       . 

-  +  -  =  0 
X       y 

(3) 

=  (l)-2 

1  =  12 

y 

(4) 

=  (3)  -  (2) 

whence 


y  =  i  and  a;  =  5 . 


110  ELIMINATION      BY      COMPARISON. 

2.  After  equation   (3)   we  may  continue  thus: 

1  1 

-  =  5 (-1)  =  (3)  transposed 

X  y 

13  11 

5 =  —  7         (5)         =  (2)  since  -  =  5 

y     y  ^  y 

whence  y  =-\  ^^^^^  ^  =  -|-' 

3.  Or,  we  may  continue   (1),  thus : 

-  =  5  -  -  (4) 

X  y 

l  =  ?-7  (5) 

X      y 

By  comparison    5 = 7  (6) 

y     y 

whence,  again  3/  =  -^  and  x  =  -|. 

In  a  similar  manner,  solve  those  equations  marked  f  in  111, 
112,  and  113. 

When  the  coefficients  and  signs  of  the  letters  x  and  y  inter- 
change, or  if  the  &igns  remain  the  same  in  the  two  equations,  we 
may  proceed  as  follows : 

Given  —  -}»»—  =  6i  (1) 

i  o 

and  vide         3.t.       4?y  _  .  .^^  ,^. 

10"y,  ex.  30        8"  "^  y  ~       ■'  ^"^ 

32x  +  21y  =  37    X  56 

21ic  +  322/  =  42^  X  56 

53aj  +  53^  =  79i  x  56 

—  llrc  +  lly=    5^x56 

a;    +    y    =     |    x  56 

—  a;4-?/    =    Jx56 

whence,     a?  =  28'    and     3/  =  56  nV/c  lOS,  ex.  1. 

In  a  similar  manner,  solve  the  equations  marked  *  in  the 
preceding  sections. 


(3) 

=  (1)  X56 

(-t) 

=  (2)  X  56 

(5) 

=  (-1)  +  (3) 

(6) 

=  (4)  -  (3) 

(') 

=  (5)  -=-  53 

(8) 

=  (6)  - 11 

THREE     EQUATIONS. 


Ill 


THREE 
INVOLVING     THREE 

115.  1.  Take  the  equations, 


EQUATIONS 

UNKNOWN     QUANTITIES. 


X 


y 


+-+-=9 


(1) 


X  z 

4  +  ^  +  8 


y 


a)  +  2y-3^=-8J 

4x  +  22/  +  2  =  36  } 
2a;  +  8y  +  ^  =  64  ]" 

—  2a;  +  6y  =  28 

12a;  +  6y  +  3^  =  108    ^ 
a;-f2y  —  3^=— 8  5 

13a;  +  8j/  =  100 

—  13a;  4-  39y  =  182 

47y  =  282 

Whence,  3/  =  6,  and  from  (6)  a;  =  4,  and  from  (4)  2;  =  8. 
Hence,  having  three  equations  with  three  unknown  quantities, 
to  find  their  values: 

(1.)  If  necessary,  clear  the  equations  of  fractions. 
(2.)  From  any  two  equations  eliminate  either  of  the  letters. 
(3.)  From  any  other  two  equations  eliminate  the  same  letter. 
(4.)  Proceed  as  in  109,  llO,  or  111  with  the  two  equations 
thus  obtained. 


(2) 

(3) 

(4) 
(5) 

=  (1)  X  4 
=  (2)  X  8 

(6) 

=  (5)  -  (4) 

(7) 
(3) 

=  (4)  X  3 

(8) 

=  (7)  +  (3) 

(9) 

=  (6)  X  ^ 

(10) 

=  (8)  +  (9) 

EXAIMPLES. 


2. 


3. 


x-\-  y  +  2  =  18 

x  +  ^i/  +  2z  =  ^S 

.+  1  +  1  =  10  J 

X  -\-  y  -\-2z=  9 
X  4-  2y  +  Sa  =  14 
6a;  -f  5?/  +  3;^  ==  25 


2z=21-i(ix-\-y) 
4.  3a;=  72 

38  =  1  (3a;  +  3^  -  ^) 


a;  +  2  (;y  +  ^)  =  31 
5.  y  +  3  (a;  +  2)  = 
2  +  4  (a;  +  y)  = 


=  31. 
=  42  > 
=  51  ) 


112 


FOUR     OR     ?.1 0  li  E     EQUATIONS 


2  +  34--^" 

8x  —  9j/  —  "iz  =  —  3G  ] 
7.    12a;  — ?/  — 3<^  =  36        ;> 

^_^  +  f=12 
3       4^2 

^ 

6a;  — 2y—  5;  =10      J 

4^2       3            J 

FOUR    OR     MORE     EQUATIONS    INVOLVING    A    LIKE     NUMBER 
OF    UNKNOWN    QUANTITIES. 


1J6.  1.  Given,   the  four  equations, 

x-\.  2y  -\-  2z  -f-   2iu  =    26 1 

ox  -\-    y    -\-  ?)Z  ■\-  3zt'  =    3G  I 

4a;  +  4y  +    z    -f  4?c7  =    44  | 

5a;  +   5 j/  +   5,^  +     zy   =    50  J 

3cc  +  63/  +  G;3  +  6w  =  78 
4a;  +  8?/  +  82  +  8z(;  =  104 
5.^  +  lOj/  +  IO2  +  lOz^  =  130 

Three  equations, 

53/+  3.-  +  Siy  =  42"] 

4y  +  7^  +  4i(;  =  60  ;> 

5^^+52+  9ii;  =  80  J 

20y  +  12^+  \2w=  168 
20y  +  35:2  +  20w;  =  300 

Two  23,2  +  8w;  =  132  ^ 

equations,      ^z  +   ^w  =    38  >" 

69^  +  2\w  =  396 
8;2  +  24t6-  =  152 


(1) 
(2) 
(3) 

(^) 

(5) 
(6) 
(7) 

(8) 

(9) 
(10) 

(11) 
(12) 

(13) 
(14) 

(15) 
(16) 

(17) 


=  (1)  X  3 
=  (1)  X  4 
=  (1)  X  5 

=  (5)  -  (2) 
=  (6)  -  (3) 
=  (7)  -  (4) 

=  (8)  X  4 
=  (9)  X  5 

=  (12)  -  (11) 
=  (10)  -  (8) 

=  (13)  X  3 
=  (14)  X  4 

=  (16)  -  (15) 


One  equation,  6I2;  =  244 

Whence,  ;2  =  4,    by  (14)  w  =  5,    by  (8)  3/  =  3,    by  (1)  a:  =  2. 

It  is  evident  that  in  the  same  way  we  may  solve  any  number 
of  equations  involving  a  number  of  unknown  quantities  equal  to 

/ 


SYMMETKICAL     EQUATIONS.  113 

that  of  the  equations,  i.  e.,  we  may  always  find  the  value  of  each 
letter  by  the  following   steps : 

(1.)  Eliminate  any  letter  by  difl'erent  combinations  of  two 
equations  till  that  letter  entirely  disappears,  leaving  the  number 
of  new  equations  less  by  one. 

(2.)  By  different  combinations  of  these  new  equations,  elimi- 
nate any  other  letter,  till  the  number  of  equations  is  less  by 
one  more. 

(3.)  Continue  these  operations  till  an  equation  is  obtained 
containing  one  unknown  quantity,  and  find  its  value. 

(4.)  Find  the  value  of  the  other  letters  by  successive  substi- 
tutions. It  is  not  nece&sary  that  every  letter  he  found  in  all  the 
equations. 


2a;  +  3?/  +  42  +  ^iv  =  40 
Zx -\- 2y  —  4.Z -{■  lu  =  —  I 
5x  4-  4y  —  2z  -{-  IV  =  11 
7x  —  5y  +  3.:;  —  tv  =      2 


EXAMPLES. 

3u-i-  X  -}-2y  —  z=22 


4:x—  7/  -\-3z         =35 

'   4u-\-3x  —  2y         =19  " 

2u  +  4.y  -\-2z         =46 


SYMMETEICAL     EQUATIONS. 

lit.  The  preceding  principles  will  solve  any  set  of  equations 
which  can  occur.  Nevertheless  there  are  many  short  processes 
which  depend  upon  the  nature  of  the  equations  involved.  Most 
of  these  processes  occur  in  connection  with  what  are  called 
symmetrical  equations,  some  examples  of  which  we  will  now  give. 

1.   Given,  to  find  x,  y,  and  z. 

'-  +  '-=  ^  (3) 

10 


114 


S  Y  .AI  M  E  T  K  I  C  A  L      EQUATIONS 


999         042 

-  +  -  +  - 
X       y 


z         i 


1^ 


1       1       1       121 

1_   49 
i~  72I1 

1_  41 

^~72U 

1_  3]_ 

z~  72U 

49  = 


Whence, 


a-.  =  14 


(i) 

=  (1)  +  (2)  +  (3) 

(5) 

=  (4)  -4-  2 

(6) 

=  (5)  -  (3) 

(') 

=  (5)  -  (2) 

(8) 

=  (5)  -  (1) 

y  = 

.17??,            .  =  23l. 

In  a  similar  manner  solve, 


5. 


x  +  y=19^ 
X  -]-  z  =  IS 
y-{-z  =  17 


1113 

-+-+-=0 
X      y       z       b 

1       11       11 

-+-+    -=9l 

X      y       to       J,4: 
1111 

X  Z  IV  ^ 

1       1        1       13 

y  Z  lU  Ji'k 


9  O 

-+- 

X       y 

2       2 

-+  - 
X         z 


i 

6 

16 

15 

2       2        9 

-+-  = 
y        z 


roj 


x+y  -\-  z 
x-\-y  -\-w 

y  +  z  -i-  IV 


X 


z  +  iv  = 


211 
22 
24 
23 


^  +  3  (y  +  ^) 

6.    ?/  +  3  (x  +  e) 

2;  -f  3  (ic  +  y) 


28 
26 


rJlOBLEMS.  115 

PROBLEMS 
INVOLVING    TWO    OR    MORE    UNKNOWN    QUANTITIES. 

118.  All  problems  must  involve  as  many  independent  equations 
as  they  contain  unknown  quantities. 

EXAMPLES. 

1.  *The   sum  of   two   numbers   is   30,  and    their  difference   6 
What  are  the  numbers?     (^Vide  108,  ex.  1.) 

2.  Three  times  the  money  of  A  added  to  twice  that  of  B 
would  make  §22 ;  but  twice  that  of  A  added  to  three  times  that 
of  B  would  make  $23.  What  is  the  amount  each  has  in  pos- 
session?    (Vide  lOS,  ex.  2.) 

3.  *If  the  first  of  two  numbers  be  multiplied  by  3  and  the 
second  by  5,  the  sum  of  the  products  will  be  165  ;  but  if  the 
fiirst  be  divided  by  4  and  the  second  by  7,  the  sum  of  the  quo- 
tients will  be  8.     What  are  the  numbers?  Ans.  20  and  21. 

4.  If  7  be  added  to  the  first  of  two  numbers,  the  sum  will 
be  three  times  the  second ;  but  if  7  be  added  to  the  second,  the 
sum  will  be  five  times  the  first.     What  are  the  numbers? 

Ans.  2  and  3. 

5.  *The  sum  of  two  numbers  is  100,  and  their  difference  20. 
What  are  the  numbers? 

6.  *The  sum  of  two  numbers  is  40,  and  the  greater  is  three 
times  the  less.     What  are  the  numbers? 

7.  Says  A  to  B,  ''Give  me  $5  and  we  shall  have  equal  sums.'* 
Now  together  they  have  $50.     How  much  does  each  possess? 

8.  If  a  $25  saddle  be  placed  on  a  horse  his  value  will  be 
twice  a  second  horse ;  but  if  the  same  saddle  be  placed  on  the 
second  horse  his  value  will  still  be  $25  less  than  the  first  horse. 
The  value  of  each  horse  is  required. 


116  PROBLEMS. 

9.  In  a  mixture  of  corn  and  wlieat  ^  the  whole  -f  5  bushels 
was  corn;  but  -^  the  whole  +10  bushels  was  wheat.  What  was 
the  quantity  of  each? 

10.  Divide  50  into  two  parts  so  that  twice  the  first  shall  be 
^  the  second. 

11.  *  Divide  72  into  two  parts  so  that  ^  the  first  and  I  the 
second  shall  be  equal. 

12.  *  Divide  36  into  three  parts  so  that  ^  the  first,  ^  the 
second,  and  ^  the  third  may  be  equal.      (^Vide  104,  ex.  10.) 

Let  X,  y,  and  z  represent  the  parts,  and  m  the  quantity  to 
which  they  are  to  be  equal  when  divided  by  2,  3,  and  4. 

Then  x  +  y  +  s  =  36  (1) 

_  X  y  z 

and  -  =  m,  -  =  m,   -  =  wz 

J  o  4 

Whence  x  =  2m,  y  =  3???,  z  =  4;/i 

Adding  which  x  -\-  y  -{■  z  ^=.  9??z  (2) 

By  comparing  (1)  and  (2),  ^m  =  36  .•.  m  =  4,  a:  =  8,  y  =  12, 
and  z  =  16. 

13.  *The  sum  of  the  first  and  second  of  three  numbers  is  11, 
of  the  first  and  third  12,  of  the  second  and  third  13.  What 
are  the  numbers?     {Vide  IIT,  ex.  2.) 

14.  *The  sum  of  the  reciprocals  of  the  first  and  second  of 
three  numbers  is  5  ;  of  the  first  and  third  7  ;  of  the  second  and 
third  8.     What  are  the  numbers"? 

15.  A  and  B  have  the  same  income;  A  saves  \  of  his  annu- 
ally ;  but  B  by  spending  $50  per  annum  more  than  A,  at  the 
end  of  six  years  finds  himself  §150  in  debt.  What  is  the  income 
of  each? 

16.  *If  8  be  added  to  the  numerator  of  a  fraction,  the  value 
of  the  fraction  will  be  2  ;  but  if  8  be  added  to  tlie  denominator, 
the  value  Avill  be  only  |.     Required  the  fraction. 


riiOBLEMS.  117 

17.  A  number  expressed  by  two  digits  is  four  times  the  sum 
of  the  digits.  If  27  be  added  to  the  number,  the  digits  will  be 
interchanged.     What  is  the  number? 

Let  X  ==   the  left  digit,  and  y  the  right. 
Then                      lOx  +  y  =  4  (x  -f  3/)  (1) 

and  10a;  +  y  +  27  =  lOy  +  a:  (2) 

Whence  10x+y  =  36  Ans.  Vide  29,  ex.   1. 

18.  A  number  is  expressed  by  three  digits.  The  middle  digit 
is  twice  the  numerical  value  of  the  left-hand  digit,  and  is  greater 
by  3  than  the  right-hand  digit.  If  99  be  subtracted  from  the 
number,  the  right  takes  the  place  of  the  left-hand  digit,  whilst 
the  middle  digit  remains  the  same.     What  is  the  number? 

19.  A  number  is  expressed  by  four  digits.  The  fourth  or  left- 
hand  digit  is  one-half  the  second.  The  first  or  right-hand  digit 
is  less  than  the  third  by  2.  The  local  value  of  the  fourth  is  50 
times  the  local  value  of  the  second.  If  909  be  subtracted  from 
the  number,  the  order  of  the  digits  is  exactly  reversed.  What 
is  the  number? 

20.  A  number  consists  of  three  figures.  The  left-hand  figure 
is  double  the  right.  The  sum  of  the  digits  is  3.  If  81  be  sub- 
tracted from  the  number,  the  left-hand  figure  is  found  in  the 
middle,  the  middle  figure  is  removed  to  the  unit's  place,  whilst 
the  unit  figure  appears  at  the  left.     What  is  the  number? 

21.  *  A  and  B  can  do  a  piece  of  work  in  8  days;  A  and  C 
in  9  days;  B  and  C  in  10  days.  In  what  time  can  each  do  the 
virork  alone?  And  in  what  time  if  all  work  together?  (Fide 
lit,  ex.  1.) 

22.  *A,  B,  and  C  can  do  a  piece  of  work  in  2=  hours;  B, 
C,  and  D  in  lii  hours;  A,  B,  and  D  in  2A  hours;  A,  C,  and 
D   in  precisely  2   hours.     In  what  time  can  each   do  the  work 


118  PEOBLEMS. 

alone  ?      In   -what    time    can    all   do    the   work   together  ?      ( Vide 
117,  ex.   5.)  Ans.  for  all  Ihr.  oGjiiin. 

23.  A  bill  of  85000  was  paid  in  eagles  and  half-eagles,  using 
of  both  kinds  560  pieces.     What  number  of  each  was  used? 

24.  Sajs  A  to  B,  ''  Ten  years  ago  I  was  three  times  as  old 
as  you  at  that  time ;  now  my  age  is  only  double  yours."  "What 
is  the  aire  of  each  1 

25.  If  5  times  A's  property  is  added  to  |  of  B's,  the  sum  will 

be  82700.     If  5  times  B's  is  added  to  |  of  A's,  the  sum  will  be 
85100.     What  is  the  property  of  each? 

26.  I  have  a  gold  and  a  silver  watch,  and  a  chain  worth  $50. 
If  the  chain  be  attached  to  the  silver  watch,  together  they  are 
worth  I  the  gold  watch.  But  when  the  chain  is  worn  with  the 
gold  watch,  they  are  worth  5  times  the  silver  Avatch.  The  value 
of  each  watch  is  required. 

27.  At  an  election  the  majority  was  80  votes.  Had  |  the 
minority  votes  and  25  votes  more  been  given  to  the  successful 
candidate,  he  would  have  received  in  all  double  the  number  of 
his  opponent.  How  many  votes  were  actually  given  to  each 
candidate  ? 

28.  The  crown  of  Hiero,  king  of  Syracuse,  weighed  20  pounds 
in  air  and  18|  pounds  in  water.  2now  19||  pounds  of  gold  weigh 
18||  pounds  in  water,  and  10^  pounds  of  silver  weigh  9^  pounds 
in  water.  How  much  gold  and  how  much  silver  did  the  crown 
contain?  Ans.  gold  1-1.77,  silver  5.23. 

29.  *If  a  grocer  mix  sherry  and  brandy  in  the  ratio  of  2  to 
1,  the  mixture  is  worth  78  shillings  per  dozen.  If  he  mix  them 
in  the  ratio  of  7  to  8,  the  mixture  is  worth  79  shillings  per 
dozen.     "What  is   the  price  per  dozen  of  eacli  kind  of  wine  ? 


P  Ji  0  B  L  E  M  S  .  1  i  D 

30.  In  a  composition  of  gunioowder  tlie  nitre  was  10  pouncis 
more  than  |-  of  the  whole,  the  sulphur  4^  pounds  less  than  i 
of  the  whole,  and  the  charcoal  2  pounds  less  than  i  the  nitre. 
"What  was  the  quantity  of  powder? 

31.  A  vintner  sold  at  one  time  20  dozen  of  port  vrine  and 
30  dozen  of  sherry  for  $;120.  At  another  time,  30  dozen  of 
port  and  25  dozen  of  sherry,  by  a  rise  of  81  each  per  dozen, 
brought  S195.     What  was  the  first  price  of  each  per  dozen? 

32.  A's  property  together  with  -i-  B"s  and  C's  is  worth  $3500; 
B's  with  ^  A's  and  C's  is  worth  $5000;  C's  with  ^  A's  and 
B's  is  worth  $5250.     AVhat  is  the  property  of  each? 

33.  On  examining  my  watch  I  find  that  2  the  time  past  noon 
is  4  of  the  time  till  midnifxht.     What  is  the  time  ? 

3-4.  A  farmer  has  30  bushels  of  oats,  at  30^  per  bushel, 
which  he  wishes  to  mix  with  corn  at  70/^  and  barley  at  90^ 
per  bushel,  making  a  mixture  of  200  bushels,  at  80^  per  bushel. 
How  much  com  and  barley  must  he  mix  with  the  oats? 

35.  A  farmer  has  86  bushels  of  w^heat,  at  4.5.  6d.  per  bushel. 
Barley,  at  3s.  per  bushel,  and  rye,  at  3^.  6d.  per  bushel,  are  to 
be  mixed  with  the  wheat  so  as  to  make  a  mixture  of  136 
bushels  worth  4s.  per  bushel.  How  much  rye  and  barley  are  to 
be  used? 

36.  A  gentleman  left  a  sum  of  money  to  be  divided  among 
four  servants.  The  share  of  the  first  was  -^  the  sum  of  the 
shares  of  the  other  three.  The  share  of  the  second  was  -^  the 
sum  of  the  other  three.  The  share  of  the  third  was  ^  the  sum 
of  the  other  three.  The  share  of  the  first  exceeded  that  of  the 
last  by  $14.  What  was  the  amount  divided  and  the  share  of 
each? 


120  PROBLEMS. 

37.  A  person  pays  at  one  time  two  creditors  f  53,  giving  to 
one  j4j  the  sum  clue  him,  and  to  the  other  $3  over  i  the  sum 
due  him.  At  anotlicr  time  he  pays  the  two  $42,  giving  the 
first  I  of  what  remains  due  him,  and  the  second  i  of  what  re- 
mains due  him.      How  much  did  he  owe  each  ? 

38.  Three  persons,  A,  B,  and  C,  have  $96  among  them.  A 
gives  to  B  and  C  as  much  as  they  already  have.  Then  B  gives 
to  A  and  C  as  much  as  they  have ;  after  which  C  gives  to  A 
and  B  as  much  as  they  then  have.  After  this  distribution  each 
has  $32.     How  much  did  each  have  at  first"? 

39.  *A  person  has  two  kinds  of  money;  it  takes  10  pieces 
of  one  to  make  a  dollar,  and  2  pieces  of  the  other  to  make  the 
same  sum.  Some  one  offers  him  a  dollar  for  6  pieces,  if  he 
could  make  the  change  even.  How  many  pieces  were  used  of 
each  kind? 

40.  *A  man  had  dimes  and  half-dollars,  and  paid  a  debt  of 
$2  with  12  pieces.     How  many  of  each  kind  did  he  use? 

41.  A  man  has  eagles  and  half-eagles,  and  pays  a  debt  of  $65 
with  8  pieces.     How  many  of  each  kind  must  be  used? 

42.  A  man  has  100  dimes  and  half-dimes,  out  of  which  he 
paid  a  debt  of  $6.15,  and  had  27  pieces  left.  How  many  dimes 
and  half-dimes  were  taken? 


LITERAL     EQUATIONS.  121 


LITERAL      EQUATIONS. 
119.  1.  Given,  the  equations, 


and 


Whence 


a       h 

-  +  -  =  w 
X       y 

(1) 

h        c 

-  +  -  =  « 

X       y 

(2) 

X       y 

(3) 

=  (1)  X  6 

ah      ac 

1 ssr  an 

X        y 

(4) 

=  (2)  X  a 

h^      ac 

y     y " 

-  an 

(5) 

=  (3)  -  (4) 

h^  - 

—  ac 

(6) 

^  -hm- 

-  an 

ac       he 

1 =  cm 

X        y 

(7) 

=  (1)  X  c 

h^       he       ^ 
X        y 

(8) 

=  (2)  X  i 

h^      ac 

X           X 

-  cm 

(9) 

=  (8)  -  (7) 

-  ac 

(10) 

Whence 

2.  Given,  ax  -{-  hy  =  m  and  ex  -\-  dy  =  n,  to  find  x  and  y. 

dm  —  hn  an  —  cm 

Ans.  X  =  — ; —  ,   y  =  — ■ -— . 

ad  —  DC  ad  —  be 

X      y 

3.  Given,   -  +  r  =  1  and  x  +  y  =  c,  to  find  x  and  y. 

.  (c  —  h)  a  (a  —  c)h 

Ans.  X  =  -^ f-  ,  y  =  ^ ^  , 

a  —  b     '  ^         a  —  b 

4.  Given,  — f-  -  =  a,    -  +  -  =  5,  &  -  H —  =  c,  to  find  x,  ik  z. 

^  X      y  X       z  y       ^ 

2  2  2 

a  +  6  — c     -^       a  — 6-fc'  __  «  4.  5  _j.  ^ 


122  0  !•.  N  E  R  A  L  I  Z  A  T  I  O  X  S  . 

112112  112 

5.  Given,   -+-  =  -,  --f-=-,  &  --f-=-.to  lliiJ  x,  y,  z, 
X       y       a     X        z       b  y        z        c 

ahc  ahc  nhc 

Ans.  X  —  - — ;  ,  y  = 


hc-\-ac  —  ah  he  —  ac-\-ah''  — he -]- uc -\- ah' 

ah  c        d 

6.  Given,   -  -X-  -  =z  m  and  -  -f  -  =  ??,   to  iincl  x  and  ?/. 

X     y  X     y 

ad  —  he  ad  —  he 

Ans.  X  = J-  ,  y  = —  . 

dm  —  bn  an  —  cm 

X  II  X'  '?/ 

7.  Given,   -  +  '^  =  m  and  -  -f  '-  =  n,  to  find  x  and  y. 

a       b  c       d 

(dii  —  hni)  ae  (am  —  ciC)  hd 

Ans.  X  =  ^ r —  ?    V  =  ^ ] r • 

ad  —  he      '   *^  ad  —  he 

8.  Given,  ax  +  hy  =  m  and  hx  -f  ay  =  ni,  to  find  x  and  y. 

m  m 

Ans.  X  =  r ,    y  ==  7  . 

a  -^r  ^  «  +  ^ 

9.  Given,  — | —  =  -   and  ax  =  hy,  to  find  x  and  y, 

X       y       c 

(a+h)  e  (a  +  h)  c 

Ans.  X  =  ^^ ,   y  =  ^ — j-^  . 

a  b 

GENERALIZATIONS. 

120,  1.  A  and  B  together  can  do  a  piece  of  work  in  c  days. 
The  time  in  whicli  A  can  do  the  work  is  to  the  time  in  whicli 
1*  can  do  it  alone  as  h  is  to  a.  In  what  time  can  each  do  it 
alone?     The  equations  of  this  problem  are  those  of  ex.  9,  119, 

-rx  v  .        («  +  ^')c  (a  +  Z))c 

Hence,  A  requires  ,  and  L  -^^ ; . 

a  b 

If  c  ==  20,  a  =1,  6  =  2,  then  A  requires  60,  and  1>  30  days. 

2.  A  person  has  two  kinds  of  money :  it  takes  a  pieces  of 
the  first  to  make  a  dollar,  and  h  pieces  of  the  second  to  make 
the  same  sum.  Some  one  offers  him  a  dollar  for  c  pieces.  How 
many  of  each  kind  did  it  take?  The  equations  of  this  problem 
arc  those  of  ex.  3,  119. 

ITencG  it  takes  ^^ f-   and   ^— ^~-  . 

a  —  h  a  —  h 


GENE  R  A  L  I  Z  A  Tl  0  X  S  .  123 

If  a=  10,   h  =  2,  c=6,  the  problem  is  39*,  IIS. 
If  a  =  G },   6  =  13,   c  =  8,   the   problem  is  41,  118. 

Heading  §05  for  a  dollar. 
If  a  =  61i,  h  =  123,  c  =  73,  the  problem  is  42,  US. 

Heading  $G.15  for  a  dollar. 

3.  If  a  grocer  mix  sherry  and  brandy,  in  tlie  ratio  of  a  to  Z>, 
the  mixture  is  worth  m  dollars  per  dozen.  If  he  mix  in  the 
ratio  of  c  to  d^  the  mixture  is  worth  n  dollars  per  dozen. 
AVhat  is  the  price  of  each  per  dozen  ? 

Sherry  Brandy 

(a  +  V)  dm  —  {c-\-  d)  hn      (c  -f  d)  an  —  (a -\- h)  cm 

Ans. T ,   — — — 7 

ad  —  be  ad  —  be 

If  a  =  2,  6  =  1,  m  =  78,  c  =  7,  d=  2,  n=  79,   the  problem 

is  29,  lis.      Vide  2S,  ex.  13. 

4.  The    sum    of   two    numbers    is    a,    and   their   difference   h. 

a  -{-  h  a  —  h 

AVhat  are  the  numbers  ?  Ans.  x  =  — ——  ,    ?/  =  — - —  . 

2  2 

If  a  =  30  and  Z>  =:  G,  the  problem  is  ex.   1,  IIS. 

If  a  =  100  and  h  =  20,  the  problem  is  ex.  5,  118. 

In  the  same  manner  generalize  the  problems  marked  *  in  IIS. 

5.  Find  X  and  y  in  the  equation, 

x-f  y  =c  (1) 

and  f+l^  =  x^-\-  «2  (2) 

^ince  X  =  c  —  y,  we  have  y^  -\-  h^  =  (^c  —  w)-  -f-  a,^  (3) 

that  is,  3/2  _|_  ^2  ^  ^2  _|_  2ry  -f  y2  4-  a2  (4)  * 

C2  -f  a2  _  ?,2  C2  +   Z>2  _  ^2 

Whence,  y  = and    x  = . 

'  •^  2c  2c 

6.  A's  property,  together  with  I  times  what  B  and  C  are 
worth,  is  equal  to  p  dollars.  B's  property,  together  with  7n  times 
what  A  and  C  are  v/orth,  is  equal  to  q  dollars.  C's  property, 
together  with  qi  times  what  A  and  B  are  worth,  is  equal  to  r 
dollars.     "What   is  the  property  of  each  ? 


124 


GENEKALIZATIOKS. 


Solution: —  ^  +  Ki/  +  ~)  =  P 

y  4-  m  ix  +  s)  =  (/ 

z  +  n  (x  -T-y)  ==  r 

X  —  Ix  -\-  Ix  -\-  ly  -\-  Iz  ■=:i  J) 

x(l-l)-\-l(x^yJ^z)=p 

X  4- {x-^y  -]-  z) 


? 


1  — / 

m 


1  — / 


(1) 
(2) 

(3) 

(4) 
(5) 

(6) 


^  +  i^(-  +  ^  +  ^~)  =  rf;;iK') 


+ 


71 


(x  +  y  +  z) 


(8) 


1    91    '^  '     ^       '         ^  2    ^^ 

•^  \1 — I     \-\-m     1 — 7i/  1 — /     1 — m      1 — n 

(9)  =(6) +  (7) +  (8) 

1 4-  ^ — ,  + . +  -. — • )  (^+y  +  ^)  =  i^+r^  +  i — 

1  —  /       1 — 771       1 — n/  1  —  /      1 — 7n      1  —  ?i 

(10)  =  (9)  factored 


P     +_!_+       '' 


'^  +  ^  +  -  = 


1  —  ?/z        1  —  n 


1  + 


Z 


1  — 

P 


X  = 


i" 


771  n 

1  771  1  71 

q  ?' 


(11) 


1  7/1  1   71 


1—  /  1   _/  / 


1   — 


771  11 

+    1 +  1 

1  771  1  71 


(12) 


q  in 

^  1   7/1  1  111   \    ^  I 

r 


+ 


+ 


1  /  1  771  1  71 


Z   = 


7«  n 

1 -/+1 +1 

L  —  t  1  —  771  1  —  71 

7)  7  7'         1 

tS  +  T^—  +  1 

1  /  1   771  1   —  11    \ 


!■  (13) 


1—71  1—71  / 

i^  +  r^ 


111  11 

4-  ; + 


(14) 


1  _  m.    '    1  —  ;/  J 


NEGATIVI^      KEfciULTS.  125 

Equation  (4)  is  obtained  by  subtracting  and  adding  Ix  from 
the  first  member  of  (1).  Equation  (12)  is  obtained  from  (6) 
by  transposition  and  the  substitution  of  the  value  of  x  -^  y  -\-  z. 
Equation  (7)  is  obtained  from  (2)  by  taking  the  same  steps  as 
were  taken  on  (1),  or,  what  is  much  better,  (7)  may  be  ob- 
tained from  (6)  by  writing  an  equation  of  the  exact  form  of 
(6),  but  using  the  next  letters  in  the  alphabet,  returning  to  the 
Jirst  letters  when  all  in  the  original  equations  are  exhausted. 
Equation  (8)  is  obtained  from  (7),  and  (13)  from  (12),  also 
(14)  from  (13)   in  the  same  way.     (^Vide  llf,   1.) 

If  /  =  2,  111  =  3.  n  =  4,  p  =  56,  q  =  77,  r  =  96,  then  x  =  10, 
^=11,  .=  12. 

If  /  =  ^,  m  =  ^,  71  =  J,  ^;  =  3500,  q  =  5000,  r  =  5250,  then 
X  =  2000,  y  =  3000,  z  =  4000.     (  Vide  118,  ex.  32.) 

NEGATIVE      EESULTS. 

121.    1.   A  wishes   to   pay   a   debt   with    100   dimes   and   half- 
dimes,   the  debt  being  $4.      How  many  of  each  are  required  ! 
Let  X  =   the  dimes,  and  y  =   the  half- dimes. 

Then  from  the  question        x  -{-  y  =  100  (1) 

and  10.T-f5y  =  400  (2) 

Whence,  x=  —  20,    and   y  ==  120. 

Erom  this  answer  we  discover  that  it  is  impossible  to  pay  $4 
by  the  use  of  exactly  100  dimes  and  half- dimes. 

A  must  pay  120  half-dimes,  and  receive  back  20  dimes  in 
change.     Had  the  question  read, 

1.*  In  paying  a  debt  of  $4,  A  gave  100  more  half-dimes  than 
he  received  dimes  in  change.  What  number  of  ejich  passed 
between  the  parties? 

We  should  then  have,  y  —  x  =  100  (1) 

and  5^-10x  =  400  (2) 

Whence,  .r  =  20,    and    .y  =  120. 


126  NEGATIVE      RESULTS. 

The  results  are  now  positive,  sliowing  the  problem  to  be  arith- 
metically possible.     (^Vide  51  et  ante.^ 

In  general,  if  in  the  solution  of  a  problem  a  negative  result 
is  obtained,  we  conclude  that  the  problem  as  enunciated  involves 
an  arithmetical  impossibility.  But  the  results,  whatever  they 
may  be,  are  interpreters  of  the  error,  and  guide  to  a  proper 
enunciation  of  the  problem. 

2.  The  sum  of  two  numbers  is  20,  and  live  times  the  one 
added  to  six  times  the  other  makes  the  sum  of  25.  What  are 
the  numbers'?  Ans.  a;  =  95,  y  =  —  75. 

The  result  shows  that  the  problem  should  have  read, 
2.*  The  difference  of  two  numbers  is  20,  and  i\YQ  times  the 
greater  diminished  by  six  times  the  less  makes  a  remainder  of 
25;  for  if  we  substitute  x  and  — y  into  the  equations  x -\- y 
s=  20  and  hx  -\-  'oy  =■  25,  they  become  x  —  ?/  =  20  and  bx  —  6y 
=  25  ;  in  the  last  two  of  which  ic  =  95   and  y  =  75. 

3.  If  1  be  added  to  the  numerator  of  a  fraction,  its  value  will 
be  E;  but  if  1  be  added  to  the  denominator,  the  value  will  be  i. 
What  is  the  fraction? 

Let  -   =   the  fraction. 

re  +  1       2  ,  X  1 

Then  =  ^         and         — TT  =  q 

y  7  y  +  1       2 

Whence  x  =  —  3  and  y  =  —  7 

3 

The  fraction  is  therefore  — — ,  which  must  not  be  interpreted 

as  — — .     The  problem  should  have  read, 

+  7 

3.*  If  1  be  suhtracied.  from  the  numerator  of  a  fraction,  its 
value  will  be  | ;  but  if  1  be  subtracted  from  the  denominator, 
the  value  will  bo   s.      Here  .r  —  3,  y—1,  and    the  fraction  is  I. 


N  E  (i  A  T  1  Y  E      11  E  )S  U  L  T  S  . 


12: 


4.  Two  trees,  a  and  h  feet  high,  are  situated  c  feet  from  each 
other  on  a  horizontal  plain.  At  what  point  between  the  trees 
inust  a  man  stand   to  be  equally  distant  from  the   top  of  each  ? 

E 


Suppose  the  point  to  be    C : 

Let  X  =   tlic  distance  from  A   to   6', 

and  y  =   the  distance  from  B  to   C. 

Now,  by  the  Pythagorean  Proposition,  Euclid,  book  I,  47, 

nC~  =  BC"  4-  BD'^   and   EC"  =  AC-  -j-  AE\ 
But  by  the  question,  DC  =  EC,  or  axiom  7,  DC^  =^  EC^. 
Therefore,  axiom  1,       BC"  -^  BD^  =  ylC^  -f  AE\ 
That  is,  y-    -f     ?>^     ==    •'^^    +    «^  (1) 

But,  rc  +  7/=    c  (2) 


Whence,  x  = 


+  1' 


cr-  -f-  a2  _  ^2 
and  y  = — .   rule  12©,  ex.  5. 


I.  If  a  =  80,   h  =  GO,  and  c  =  100,   then  x  =  36,  y  =  04.    ) 

Vide  104,  ex.  14.j" 

II.  If  a  =  100,  h  =  40,  and  c  =  50,  tlien  x  =  —  59,  y  =  109. 

This  shows  that  with  these  values  of  a,  h,  and  c  it  would  be 
impossible  to  stand  between  the  trees  and  be  equally  distant  from 
the  top  of  each.  In  this  case  the  value  of  x  must  be  taken  to 
the  left  of  A  on  the  prolongation  of  BA,  as  in  the  following 
fiuurc. 


128 


NEGATIVE     EESULTS 


In  general,  then,  if  lines  to  the  rigid  of  a  point  are  considered 
positive,  those  to  the  left  will  be  negative.  How  should  the  prob- 
lem have  read? 

III.  If  a  =  40,  5  ==100,  and  c  =  50,  then  a;  =109  and  ^  = 
—  59. 

This  shows  an  impossibility  similar  to  the  last.  Tiiis  case 
causes  y  to  be  taken  to  the  right  of  the  point  B,  on  the  pro- 
longation of  A  B,  as  shown  by  the  figure : 


If,  then,  lines  to  the  left  of  a  point  are  considered  positive, 
those  to  the  right  will  be  negative.  How  should  the  problem 
have  read? 

Any  supposition  which  makes  x  ot  y  negative  must  be  inter- 
preted in  like  manner. 


NEGATIVE     RESULTS. 


129 


122.         INTEEPKETATIOIS"     OF     THE     SYMBOLS, 
0  J.  ^ 

loo 

where  A  represents  any  finite  quantity. 

The  formulae  of  the  last  problem  are, 

c2  +  6=  —  a2  c^  _|.  a2  —  h^ 

X  =  ^ and    y  =  - 


2c 


•2/ 


I.   If  a  =  100,  h  =  80,   c  =  60,  then   the  formulae  reduce   to, 
3600  +  6400  —  1000  0 


X   = 


120 


120 

y  = 


and 

10000  +  3600  —  6400 
l20 


=  60 


This  value   of  y   shows   that   the   point   is   at   the   foot   of  the 

0 
taller   tree,  and   therefore   the   expression   — —  must   be   infinitely 

small.     This  is  illustrated  by  the  figure. 


E 


D 


c=60 


A 


B 


Any  supposition  which   makes  (? -\- l/^  =  cv^  must   make  y  =  c, 
for  if.  in  the  value  of  y  above,  we  should  insert  c'  -f  h^  for  a^, 


we  have, 


y  = 


C2  4_  c2  -f  Z)2  _  52 


0 


Zc 


Hence,  we  assume  that  —  is   the   representative   of  an   infinitely 
small  quantity;  that  is,  the  following  equation  is  true. 

^  =  0  (1) 


130 


NEGATIVE     K  E  S  U  L  T  S  . 


II.  If  a  =  100,  h  =  80,  c  =  0,  then  the  formuhie  reduce  to, 
0  +  6400  —  10000        —  3600 


X  = 


0 


0 


y 


md 

0  +  10000  —  G400        3600 


0 


0 


By  the  supposition  the  tree  A  occupies  the  same  spot  with 
the  tree  J5,  differing  only  in  height.  It  is  clear  that  as  the 
point  C  recedes  from  A,  the  distance  to  the  top  of  each  tree 
approaches  nearer  an  equality,  and  the  two  distances  are  abso- 
lutely equal  only  when  C  is  infinitely  removed  from  A.    Hence 

—  3600         3600  ,      .  .  .   , 

the   expression —  or  must   be  mjimtehj  great^  as  each 

0  0 

of  them  is  the  representative  of  the  same  distance  en   different 
sides  of  the  point  A.     The  supposition  of  c  =  0  and  a>»  or  <Ch 

may  always  be  reasoned  upon  in  the  same  way.     Hence  —  rep- 
resents an  infinitely  large  quantity,  that  is. 


A 

0=^ 


(2) 


III.  If  a  =  80,  h  =  80,  c  =  0,  the  formulae  reduce  to. 


X  = 


0  -f  6100  —  6400 


and      1/  = 


0  -f  6400  —  6400 


0  0       ■"'     -^  0  0 

By  the  supposition  the  trees  must  be  absolutely  identical,  and 
it  is  clear  that  the  point  C  may  be  any  where  on  the  line  pass- 
ing through  its  base.     Hence  the  symbol  is  one  of  indctrrh/diaiiou, 

^  =  1,   2,   3,   -1,  -2,   -X   t.    1     &e.         GO 


That  is. 


0 


NEGATIVE      liESULTS.  131 

123.  1.  A  boy  bought  apples  and  oranges,  giving  3  cents  each 

for  apples  and  4  cents  each  for  oranges.     How  many  can  he  buy 

for  100  cents? 

Solution. 

From  the  question  we  can  have  only  one  equation,  viz : 

3a;  +  %  =  100  (1) 

Hence,  x  =  ^, — ^  (2) 

Place  y  for  x  in  (1),  and  we  have, 

100  —  4y  -f  4y  =  100  (3) 

Then,  (4  -  4)  y  =  100  -  100  (4) 

y  =  ,-  (^) 

Hence,  the  boy  may  purchace  any  number  of  either  he  pleases. 
If,  however,  none  of  the  fruit  is  to  be  cut,  the  limitation  gives 
rise  to  one  of  the  most  interesting  departments  of  algebra,  known 
as  Indeterminate    Analysis. 

EXAMrLES. 

2.  Find  x  and  y  in  whole  numbers  in   the  equation  ox  -{■  4:y 

=  100. 

100  —  4?/        .^^  ,     1—y 

We  have,      x  =  ■ — ^  ==  33  —  ?/  H ^  . 

o  o 

1—y 
Since    33  —  y    is    a   whole    number,   — - —   must   be   a   wliole 

number  also. 

1  —y 

Let  ^  = ;)  • 

then  y  =  1  —  Sp  1) 

and  o^  =  33  -  (1  -  3^;)  +  P  =  33  +  4/;  -  1 

or  .X  =  32  +  ^P  (2) 

Then  j?  may  be  any  whole  number  whatever  that  will  render 
X  and   ?y  positive  in   equations  (1)  and  (2).     It  is  evident  from 


132  NEGATIVE     EESULTS. 

(1)  that  21  must  be  0,  or  negative.     It  is  evident  from   (2)   tliut 
p  must  be  the  following  quantities,  viz  : 

0,    -1,    -2,    -3,    -4,    -5,    -6,    -7,    -8. 

Whence,      x  =  32,      28,      24,      20,      16,      12,        8,        4,      0. 

3/  =    1,       4,        7,      10,      13,     16,      19,     22,     25. 

Hence,  the  boy  in  example  1   could  have  bought  any  of  these 

combinations  of  apples  and  oranges. 

3,  Find  X  and  y  in  whole  numbers  in  the  equation, 
llx  -f  5^  =  254 

Alls.  X  =  19,    14,    9,    4. 
y  =  9,    20,    31,    42. 

As   this   subject   is   not    strictly   elementary    in    its  nature,    v/e 
shall  not  pursue  it  farther. 


r 


CHAPTER    YI. 

ii;rvoLiiTio]^. 

124. 

(1.)  The  first  power  of  a  quantity  is  the  quantity  itself. 

(2.)  The  second  power  of  a  quantity  is  the  quantity  multiplied 
by  itself. 

(3.)  The  third  power  of  a  quantity  is  the  quantity  multiplied 
by  the  2iid  power. 

(-i.)  The  m'^  power  of  a  quantity  is  the  quantity  multiplied 
by  tlie  (m — 1)'*  power. 

(5.)  Involution  investigates  the  method  of  finding  any  power 
of  a  quantity. 

(6.)  If  we  multiply  3a  by  3a  we  have  9a^ ;  and  4a2  multi- 
plied by  4a2  gives  16a^  and  7x'  X  7x'  X  Ta:^  =  349a;9  =  the 
cube  of  Tx'.     Hence, 

125.   To  raise  a  positive  monomial  to  any  required  power : 
Haise  the  coefficient  to  the  required  power  hy  multiplication,  and 
multiply  the  exponents  of  the  letters   by  the  number  expressing 
the  power  to  be  obtained. 

EXAMPLES. 

1.  Find  the  2nd  power  of  3a^  Ans.  9cf^ 

2.  Find  the  ^rd  power  of  4a*.  Ans.  64a'^ 

3.  Find  the  ^th  power  of  2x'^y.  Ans.  Q-ix^^y\ 

4.  Find  the  %th  power  of  Zx^y*.  Ans.  6561a;'«3^"^ 


134  INVOLUTION. 

5.  Find  the  squares  of  2x.  ox^y,  4;rj/-,  6x^1/^,  ^^^y^  ^^y^i  ^"^^ 
Sx^y*. 

6.  Find  the  cubes  of  2x^,  ^^y^,  ^^^y-i  8^?'^',  9.77y^,  10;r^^,  and 
\2x^ifz^. 

7.  If  we  multiply  — x  by  — x  the  product  is  -f^^  ^»<i  if  "vve 
again  multiply  +^^  by  — a;,  the  product  is  — x^  and  — x^  X 
— X  =  4-*'^^  &c.  But  — X  =  Ist  power  of  — x,  and  4-^^  == 
2nd  power  of  — x,  and  — x^  =  3rd  power  of  — x,  and  -\-x*  = 
4th  power  of  — x.     Plence, 

126.  The  odd  powers  of  a  negative  monomial  are  negative,  the 
even  powers  positive. 

EXAMPLES. 

1.  Find  the  3i'd  power  of  —  4a^x.  Ans.   —  G4a^x^. 

2.  Find  the  2nd  power  of  —  16xy.  Ans.  25  6x^^^ 

3.  Find  the  2nd  power  of  —  6x^9/^.  Ans.  2^x'^y'^. 

4.  Find  the  cube  of  —  ^x^y.  Ans.  —  126x^y^. 

5.  Find  the  14ih  power  of  —  axy.  Ans.  a^^x^^y^*. 

6.  Find  the  9th  power  of  —  x^y*.  Ans.  —  x^^y^^, 

7.  Find  the  2nd  power  of  2x^y^.  Ans.  4xy. 

8.  Find  the  3rd  power  of  3x*y'.  Ans.  27xy'^. 

9.  Find  the   4ith  power  of  ^x*y'^.  Ans.  Q2^xy^, 

J.    i_  11 

10.  Find  the  bth  power  of  —  4x"^^^  Ans.  —  1024x'?/*. 

11.  Find  the  10^^  power  of  —  2x'''y'^\  Ans.  1024a;'°°y. 

I   .^t  49 

12.  Find  the  1th  power  of  —2x^y  \  Ans.  —  l2Qxy^, 

\  1 

13.  Find  the  Gth  power  of  —  2x^y^.  Ans.  Q4:xy. 

14.  Find  the  4th  power  of  2^xy.  Ans.  390625rcV- 

15.  Find  the   squares  of   —  x^,  x^y,   Sxj/^,    —  4ic^y,    5a:^',   x^y^, 

x^y^,  and  .x*. 

1         1  1  2 

16.  Find  the  cubes  of  —  x^,   0^*3/*,  S.x^?/',    —  x^y"^,   —  7xy^,   Sxy, 

and   —  x''* 

17.  Find   {lie  2nd  power  of  o^.  Avs.  a"^'. 


INVOLUTION.  13i> 

18.  Find  the  m'^  power  of  a"^.  Ans,  a'"". 

X 

19.  Find  the  I  power  of  a'.     {Vide  §13,)  Ans,  a^. 

1  X 

20.  Find  the  mth  power  of  a*.     {Vule  §13.)  Ans.  a^n. 

If  a  =  16,  X  =  3,  m  ==  4,  then  a'"  =  16 *  =  8. 

IST.  To  raise  a  monomial  fraction  to  any  required  power : 

Observe  if  the  fraction  is  in  its  lowest  terms;   if  not,  reduce  it, 

and  then  raise   the  numerator  and  denoininator  separately  to 

the  required  power. 

32jc^V^  y  V^ 

1.  Find  the  hth  power  of         /    =  ^  Ans.     ^ 


Ux'y^        2x^  32x- 


x^y^  y^* 


2.   Find  the  StJi  power  of  —~. —  .  Ans.      . 

x^y  x^ 


x^y  —  x^ 


3.  Find  the  5^/^  power  of —  .  Ans.      ^^ 

—  'Ixy  6'1 


7 


—  ^  %^  X        x'^  8x*  V 

4.  Find  the  4ith  powers  of  — ^^  , ,  — - ,  and  j . 

—  Sx^         2y    2x^  4.7-j/' 

^    -r..    -.    1       .  7  z'     ^        ITx^y     —  x^         ,  44x^ 

5.  rind  the  bin  powers  oi   -,  ,  j,  and  j.' 


_  ys      Mxy      —  y'  boy^ 


■y 

a^  a^ 

6.  Find  the  2nd  power  of  —  .  Ans.  — r- . 

7.  Find  the  ra^^  power  of  — .  Ans.  — . 

12S.   To  raise    the   positive  binomial   x  -\-  y  to   any  required 
power : 

Multiply  the  binomial  as  indicated  in  §124. 

« 
1.  Find  the  2nd,  3rd,  4:th,  &c.  powers  of  a;  +  y. 


136 


INVOLUTION". 


Solution. 


Multiply 
by 


x-\ry 


^  \&t  power. 

~-?   (IX-l,  I.) 


x^-\-    xy 

xy  -f  3/' 

and  we  have 

x'^+  2xy  4-  y^ 

Multiply  the  1 
^nd  power  by  \ 

x-\-y 

x^-\-  2x^y-\-  xy^ 

xhj-\-   2xy^  +  y^ 

and  we  have 

X^+  3.T=j/+     3.TJ/2    +     7/3 

Multiply  the  } 
3ri  power  by  \ 

x-\-y 

x'^-^-  ox^y-\-   'ix'^y^  +   ^U^ 

x'y-\-   3a^y-f   Zxy'  +  y' 

and  we  have 

x^4-  4:x^y-\-  Qx^y^  +  "^^^^  +  y* 

Multiply  the  ? 
^th  power  by  \ 

x-\-y 

x^-\-  4:X^y-\-   Qx^y"^-  -\-  ^x^y^  -\-  xy* 

x'^y-\-  4iX^y^  +   Qx^y^  -f  4iXy*  +  y^ 

''2nd  power. 
(124,  2.) 


'ird    power. 
(124,  3) 


5  ^^^  power. 

—  I   (124,  4.) 


and  we  have  x^-{-  5x^j/+10xy  +10xy  -f  ^xy*  -\-  y^  =  ^th  power. 

1.  By  the  mere  observation  of  either  of  the  above  powers,  the 
law  which  governs  the  exponents  of  both  letters  is  readily  dis- 
covered. 

2^he   exponents   of  the  Jirst   and   last  terms   are  indicated   hy  the 

rOWER    ITSELF. 

The  exponents  of  x  decrease  towards  the  right  by  unity. 

The  exponents  of  y  increase  towards  the  right  by  unity. 

Therefore  x  disappears  from  the  last  term,  and  y  cannot  be 
found  in  the  first.      (^Vide  12^  3.) 

Disregarding  the  coetiicients,  the  several  powers  of  .t  -}-  y  may 
easily  be  written  thus : 


INVOLUTION.  137 

X  -\-  y  =  Id   puwer. 

a-?  +  ^'y  -\-  y^  =  2//(Z  power. 

x^  +  x'^u  -f  xy^  -\-  y^  =  2n-d  power. 

a;"  +  x^y  +  x^y^  -\-  xy^  -^  y^  =  4:th  power. 

x^  4-  ^V  +  ^^y"^  -\-  'x^y^  +  ^y^  +  3/*  =  5^A  power. 

x^  -\-  x^y  +  a^^j/^  4-  x^y^  -}-  .T^j/''  +  xy^  -\-  y^  =  6^/^  power. 

II.  When   any   coefficient   is   given,  the   coefficient   of   tlie    next 
term  can  be  found   by  the  following 

K  u  L  E . 
Multiply  the  given  coefficient   by  the  exjwnent  of  the   leading   letter 
ill   the  same   term,  and  dinde   the  product   by   the  number  exp)ressing 
the  2)lace  of  the  term  counting  from  the  left. 

2.  Let  us  insert  the  coeHicients  of  tlie  fifth  power. 

The   coefficient  of  the  first  term  must   always  be  one,  and  we 

(1) 
write  down  x^  as  this  first  term. 

1x5 
By  the  rule,  the  coefficient  of  the  2nd  term  is  — - —  =  5,  and 

(1)  (2)  1 

we  write  x^  -\-  bx^y. 

5x4 

Again,  by  the   rule,  the   coefficient   of  the  SrcZ   term  is   — ^ — 

(1)         J2)  (3) 

=  10,  and  we  write     .x*  -{-  hx'^y  -j-  l^x^y"^. 

Again,  the  coefficient  of  the  4:th  term  is  — - —  =  10,  and  we 

«j 

(1)  (2)  (3)  (4) 

write  x*  -f  bx^y  +  Kix^y"^  +  \^x''y\ 

10  X  2 
The   coefficient  of  the  bth   term  is   • =  5,  and  we  write 

(1)  (2)  (3)  (4)  (5) 

x''  +  bx^y  +  10x3j/2  +  l{}xY  +  ^xy\ 

5x1 
Finally,  the  coefficient  of  the  Qth  term  is  — - —  =  1,  and  we 

o 

(1)  (2)  (3)  (4)  (5)  (6) 

write  x""  +  5xV  +  lOx^j/^  +  \i)xY  +  bxy^  +  y\ 

which    corresponds   with    the   bth    power   as    obtained    by   actual 

multiplication. 

12 


138  INVOLUTION. 

In    the  same   niiinner   the   coelliclenls  of   the  i!>tii  power  are, 

(1)  (2)  (3)  (4)  (5)  (6)  (7) 

1x6      6x5      15  X  4      20  X  3      15  x  2      6  x  1 
^'     ~T~'    "~2~'    ~3~'     ~~r"'         5      '    "~6~'       '^    '"' 
1,         6,  15,  20,  15,  6,  1, 

and  v/hen  inserted  Avith   the  letters,  -we  have, 
(cc  4-  y)^  =  a;«  +  6.xV/  +  15a:y  +  2Qx^f  -f-  15xy  -f  6xj/*  +  7j\ 

3.  Find  the  1th  power  of  x  +  y. 

Ans.  x'  +  7ic«y  4-  21x^^2  +  35xy  +  350:^^/^  +  21arj/^  +  To-/  +  y\ 

4.  Find  the  2)id  power  of  x-{-y.  (§62.)   Ans.  x?  ■\- %xy -\- y"^ , 

5.  Find  the  3?-cZ  power  of  cc  +  ^. 

6.  Find  the  ^tli  power  of  cc  +  y. 

7.  Find  the  8<//,  9^/^,  and  \Wi  powers  of  a;  +  y. 

8.  Find  the  m^^  power  Q)i  x  -\-  y. 

m  (m  —  1) 
Ans.   (x  4-  y)"*  =  x"'  +  mx"'-^y  -\ ^- — - — -  x'^'-^y^  + 

y,,  (rn  -  1)  (m  -  2)  m  (m-1)  0/^  -  2)  (m  -  3) 

1,  z,  d,  1,  ^,  o,  % 

129.  To  raise  the  residual  monomial  (x  —  y)  to  any  required 
power : 

Make  the  odd  terms  positive  and  the  eveii  terms  negative. 

1.  Find  the  ^th  power  of  x — y. 

(1)  (2)  (3)  (4)  (5)  (6) 

Ans.  x^  —  bx^y  -\-  IQx^y^  —  lO.x^j/^  +  ^xy^  —  yK 

2.  Find  the  4:th  power  of  x  —  y. 

Ans.  x^  —  4:X^y  +  'ox?y-  —  4txy^  -\-  y^. 

3.  Find  the  2nd,  3/c?,  Gth,  7th,  Sth,  dth,  and   lO^A  powers  of 

x  —  y. 

4    Find  the  m"^  power  of  (x — y). 

'"  ('>>f  —  1) 
'A?is.   (x  —  ?/)"'  =  ^"^  --  mx'^'-^y  4 ^- — - — -  x'^-Y  — 

I,  -J, 

,^  („i  _  1)  (m  -  2)        ,3    ,    jn(m~l)(m-2)(m-3)    „,  .  ,    , 

1,     *j,     O,  J)     ->     «^i      i^. 


INVOLUTIOX.  139 

130*   To  raise  arnj  binomial    to  any  required  power : 

Consider  each   term  of  the   binomial  as  a  single   exjn-ession,  and 

proceed  as  in  §128   or  §129.  Then   reduce   the  result  as 
indicated  hij  the  signs. 

1.  Find  the  Vith  power  of  ?>x  -f-  2j/. 

In  the  first  phice,  (3x+  2j/)'  =   (Sx)^  +  5  (Sa^)^  (2y)  + 

10  C^xf  (2yf  +  10  {?>xy  Qljjy  +  5  (3:.)  (2j/)^  +  (2^)-\ 

This,  reduced,  gives,  (3.x  +  2j/)- ==  2-43.X*  +  81 0;/;"^/  + 
lOSOxy  +        720xy       -f     24.{)xf       +32/,^ 

2.  Find  the  Zrd  power  of  2x  -f  ^y. 

First,  (2x  +  3^)^  =  {2xy  +  3  (2x)=  (3^)  -j-  3  (2x)  (3j/)=  +  (3y)', 
which,  on  reduction,  give?, 

(2x  +  3^)5=   8^3   +        36a;2y         -f       54.r3'2       +    27/. 

3.  Find  the  Wt  power  of  x  —  2y. 

First  (X  -  2yy  =  re"  -  Ax'  (2y)  +  6.x^  (27/)^  _  4.t  (2^)-^  +  (2^)^ 
^'^\^}l'^H(x  —  2yy=x^—     8x'y    +    24.ry    —     32.ry   +  1%*. 

4.  Find  the  cube  of  2x'^  —  oyK 

First,  (i2x'-Sy'y=(2xy-S(2xy(^y')-^3(2x')(3yy-iSyy, 
a^d^this  >  ^2.x2— 33/2)3=   8x^   —      36xy       +      54xy      —277/9. 

5.  Develop  (1  +  ^)^- 

J.71S.  l  +  9x+36x2H-84x2+  126rc''+  126a;^-f  84^«4-  3Ga;^  +  9a;''+  a;». 

6.  Develop  (x  +  1)*.  Ans.  x^  +  9x8  ^  ^q-^j^  ^^^ 

7.  Develop   (1  —  .x)\  ^«5.   1  —  4.x  +  6a;2  —  4x2  _^  ^4^ 

8.  Develop  (x  —  1)*.    Ans.  x^  —  5x<  +  lOx^  —  lOx^  +  5x  —  1. 

9.  Develop  {l-x^,  (l-x^)^  (xH/)S  {x'-{-yy.  (2a:+2/)^ 
and  (3x2-2?/)\ 

10.  Develop   (3x— y)^    (5ic  +  23/)=,  (1  +  «^')',  {^'  —  ^\   J^nd 

11.  Develop   (x^  +  y^y.  Ans.  x^  -f  3x^^  -f  3x5y  +  y\ 

12.  Develop  (ic^  — ^^)''.      .4/^s.  x^  —  4x^^2  ^  Oxy  —  4x%^  +  /. 

13.  Develop  (x^  —  ?/-)^  ^tw.  x  —  Sx^  +  3x^^'  —  ;y. 


140  INVOLUTION. 

131.  The  two  formulae,  §128,  8  and  §129,  4,  may  be  writ- 
ten together,  giving  expression  to  the 

BINOMIAL   THEOREM. 

m  (m  —  1)       ^  ^   ,    "in  (m  —  1)  (m  —  2) 
(x.  =b  7jY  =  a;-  ±  jnx^'-'y  +      ^^   .^     ^  x'^'-y  ±  — ^ l^t 

,  ,        m  (m  —  1)  (m  —  2)  (m  —  3)        .  .     „ 
x^f  +  -^ 1.2,?. A    ^^^'  ^''' 

If  in  this  equation  x=l  and  ?/  =  1,  we  have, 

.    ,    ..  .    .  in  (m  —  1)    ,    7)1  Cm  —  1)  (in  —  2) 

(1  ±  1)"  =  1  ±  «  + -A^  ± -^-^-i^-^  + 

m  (m  —  1)  (in  —  2)  Cm  —  3)         .    ^ 

— ^ —^ —^ .      (a.)     I.  e. 

1.2.3.4  ^    ^ 

The  sum  of  the  coefficients  of  an?/  power  of  (a^  +  y)  is  the 
same  as  2  raised  to  the  same  power,  bj  observing  the  upper  signs. 

If  now  7/1  =  3,  and  we  observe  the  up2^e7'  signs,  the  formula 
becomes,  (1  +  1)'  =  (2)^  =  1  -f-  3  -f  3  +  1,  i.  e.  the  sum  of  the 
coefficients  of  (x  +  t/Y  =  2^  =  8. 

If  m  =  4,  then  (1  +  1)^  =  2^  =  1  -f  4  +  6  +  4  +  1,  which  are 
the  coefficients  of  {x  -f  t/Y  =16. 

If  711  =  5,  then  (1  +  1)^  =  2^  =  1  H-  5  +  10  +  10  +  5  +  1, 
tlic  coefficients  of  (x  -f  i/Y  =  32. 

If  m  =  10,  then  (1  +  1)^^  =  2'°  =  1  +  10  -f  45  +  120  +  210 
A-  252  +  210  +  120  +  45  +  10  +  1. 

If  m  =  3,  and  we  observe  the  lowe7^  signs,  the  formula  becomes, 
n  —  1)'  =  1 —  3  +  3  —  1 ;  {.  e.  the  positive  terms  exactly  cancel 
the  negative,  and  in  general,  from  formula  (a),  above,  the  sum 
of  the  positive  coefficients  must  be  exactly  the  same  as  the  sum  of  the 
•negative  in  any  power  of  x  —  y. 

We  see  also,  that  in  calculating  the  binomial  coefficients^  ive  need 
io  actually  perform  the  operations  only  half  way^  hy  taking  the  first 
half  hi  the  reverse  order. 

The  formula  above  may  be  used  to  solve  any  example. 


INVOLUTION.  141 

132 -(1).  To  find  the  cube  of  a  trinomial,    {vide  65): 

Take  the  trinomial  x  -\-  i/  -\-  z.  Consider  llie  terms  x  -\-  7/  as  a 
single  expression,  and  we  may  write  the  whole  thus,  ((a^  +  y)  -}- zy. 

This  developed  as  already  explained,  gives, 

Qx  +  y)  +  zy  =  (x  +  3/)'  +  3  (x  +  y)2^  +  3  (^x  +  i/)  z' -j-  z\ 
which,  reduced,  gives 

(x  -\-  y  -\-  zy  =  x^  -f-  'ix^y  -f  ?,xy^  +  3/'  +  S.r^.^  -f  ^xyz  -f  ?>y'^z  + 
?>xz^  +  83/^2  +  2». 

This  may  be  arranged  thus, 
(x  +  3/  +  2;)3  =  a:^  +  3/'  +  -'  +  S^i'^J/  +  ^x^^  +  B^^t  ^  3^2.  _^  3^.2^ 
+  3x;2y  +  Qxyz. 

Hence,  to  cube  a  trinomial : 

Cube  the  three  terms,  and  to  their  sum  add  three  times  the  second 
2)oiver  of  each  term  into  the  first  jJOiver  of  each  of  the  others,  and 
also  add  six  times  the  product  of  all  the  terms. 

1.  Develop   (x  —  y  -\-  zy,      Ans,  x^  —  y^  -\-  z^  —  Sx^y  +  ox^z  -} 
3y^x  +  Syh  +  ^^^x  —  ^z^y  —  Gxyz, 

2.  Develop  (x  —  2y  -\-  ozy,  Ans,  x^  —  83/'  +  27z}  —  Ax^y  + 
6xh  +  83^2^  +  243/^2  +  ISzlv  —  36z^y  —  36xyz. 

3.  Develop  (x^  -i-  x  +  1)'.  Ans.  x^  +  'Sx'  +  6x^  +  7x^  +  Qx^ 
•f  Sx  +  1. 

4.  Develop  (py^  —  x  —  1)'.     Ans.     x^  —  ox^  +  ^x^  —  3x  —  1. 

5.  Develop  (x'  —  x""  —  l)^     Ans.  x''  —  Sx'°  +  5:^^  —  3x'-  —  1. 

6.  Develop    (x^  -f?/^  +  z^y,   (x^-\-  x  +  x^y  and   (1  —  x  +  x^y. 

1  5       9x-^  3       9a; 

7.  Develop   (a-  —  x^  +  J)^        Ans.  x^  —  3x^  + 4:r^  +  — 

2  -i 

3x'       , 

-X  +  4- 

132- (2).  Useful  changes  in  the  form  of  expressions. 
l^  .j,2_^f=(x-\-yy-  2xy,  or  x' -{- y' =  (x  -  yy  +  2xy. 

2.  x^  +  3/'  =  G'c  +  yy  —  3.T3/  (a?  -f  3/).  F/cZe  §  75. 

3,  ^3  _  ^3  _  (^,y  —  3^)3  —  3.7^  (a:  —  ^).  F?V/e  §  75. 


142  LOG  A  p.  IT  II  MS. 

4.  x'  +  f  =  (x  +  T/Y  —  4.rj/  (a;  +  t/)^  +  2x^f.       Vide  §  15,   §  tG. 

5.  rc^  +  ^'  =  (.« +  ^)'  —  5^^  0^  +  ;y)^  +  5»y  {^  +  y). 

6.  x^  —  y^  =  (-^  —  J/)^  +  S.tJj/  (a;  —  ?/)'  -f-  5a:^j/^  (cc  —  ?/). 

8.  a;'  +  7/5  —  a:^'  —  x^i/  =  (;x  +  ?/)  (.-c  —  ?/)l 

132 -(3).   To   find  a  term  which  will  make  an   expression  of 

the  form  x^  =b  2ax  a  j^erfect  square. 

"We  have  (x  zhay  =  x^  dz  2ax  -}-  a^,  v/here  a~  =  ^  (2a)  squared. 

Hence,    Square    half  the    coefficient  of  the  second    term,  add    tJie 
result  to  the  eorpression,  and  it  ivill  be  a  perfect  square. 

1.  Make  x^  -\-  2x  a  perfect  square.  Ans.  x"^  -\-  2x  -\-  1, 

2.  Make  x^  +  Ax  a  perfect  square.  Ans.  x^  -f-  4x  -f  4. 

3.  Make  x^  —  Gx  a  perfect   square.  Ans.  x^  —  6^4-  9. 

4.  Make  x^  —  ox  a  perfect  square.  Ans.  x^  —  3a;  +  £  . 

5.  Make  x"^ -}- 2px  a  perfect  square.  Aiis.  x^ -\- 2px -{- jp' 

6.  Make  x^  -  J^,  a  perfect  square         ^^^^^^  ^,^, 

Ans.  x^  — +  -J-- — -  . 


tr 


LOGARITHMS. 

133.  A  logarithm  is  a  number  expressing  the  power  to  which 
a  given  number  is  to  be  raised  to  produce  another  given  number. 

134.  The  number  to  be  raised  is  called  the  hose  of  the  system 
of  logarithms  to  which  it  gives  rise. 

135.  In  the  equations, 

3°=1,  3'=3,  32=9,  3»=27,  3^=81,  3^=243,  3«=729,  &c., 
three  is  the  base,  and  0,  1,  2,  3,  4,  5,  6,  are  respectively 
the  logarithms  of  1,  3,  9,  27,  81,  243,  729. 

130.  The  base  of  the  Cor.rviox  System  is  10,  from  AAhich  we 
readily  form  this  table. 


LOG  A  KIT  II  3[  8  .  lA'o 

108=         100000000 
10»    =      1000000000 
101°  ^    10000000000 
10"  =  100000000000     &c. 
Since  a"  =  1,  the  logarithm  of  1   in  a^i/  system  is  0. 

1S1[,  Any  one  of  these  equations  "|       _^  ,^ 

is  expressed  generally  by  j 

And  any  other  by  «^  =  iV 


10°  = 

1 

10^  =    10000 

10'-  = 

10 

10*  =   100000 

10^  = 

100 

10«  =  1000000 

10'  = 

1000 

10^  =  10000000 

And  by  multiplying  we  have  a"^^  =  31  X  iV.     That  is, 

The   logarithm   of  the  product  of  two  numhers  is  equal  to  the  sum 

of  their  logarithms. 

Thus,  by  the  table  above,  the  logarithm  of      10000        is      4, 

and  the  logarithm  of  100000       is      5, 

and  we  find  the  logarithm  of    ")     .  -,  AAnAnAAnn   +     i      o 

^,  1     .     ^  xi  t  >•  VIZ :  1000000000  to  be  i>, 

the  product  or  these  numbers,  j 

which  is  5  +  4. 

138.  Again,  a*  =  M 

and  ay  =  N 


M 

By  dividinfr  we  have  a"^"  =  -— ,  That  is. 

The  logarithm  of   tlie    quotient    of  two    numhers   is   equal   to    the 
difference  of  their  logarithms. 

Thus,  the  logarithm  of  100000000000  is  11. 

The  logarithm  of  100000000  is     8. 

And  the  logarithm  of  quotient,  viz:  1000  is     o, 

Avhich  is  11  —  8, 

139 -(1).  By  §126,  ex.  18,  the  m*^  power  of  the  equation, 

a'  =  M 
is  a*^  =  il/'".  That  is, 

Tlic   logarithm  of  the  w**  jiower  of  a  numher  i-^  ri   times  the  log- 
arithii  of  the  numher. 


144  L  U  G  A  ]l  1  T  H  M  S  . 

Thus,  the  logarilliin  of  100  is     2, 

and  the  lofiarithin   of    ")      .  i  Aa/vnAAAnnA   •     -<  a 

^i      p.,,       "^         ^-.^n    /-viz:  10000000000  is  10, 

the  bill  power  oi  100,  j  ' 

which  is  5  X  2 
139- (2).  By  §12G,  ex.  20,  the  m'*  root  of  the  equation, 

a*  =  M 


« 


ni. 


is  a"*  =  i/M,  That  is. 

The   logarithm  of  the   in*'^  root   of  a   number   is  the  logarithm   of 

the  number  divided  by  m. 

Thus,"  the  logarithm  of  10000000000  is  10, 

and  the  logarithm  of  the  ")     .  i  nn   •       o 

^th  root  of  10000000000,  j"  ^'"^ '  ^^^  '^     "' 

which  is  10  -T-  5. 

140.   By  examining  the  table  §  136,   we  see  that. 

The  logarithms  of  numbers  between  1  and  10  must  be  greater 
than  0  and  less  than  1. 

The  logarithms  of  numbers  between  10  and  100  must  be 
greater  than   1  and  less  than  2. 

The  logarithms  of  numbers  between  100  and  1000  must  be 
greater  than  2  and  less  than  3,  &c. 

Tlie  following  table  illustrates  this,  Avhere  the  decimals  are 
carried  to  6  phices. 

10^  =1         lO-'^^^^'"     =       5        lO'-''^^'^!^  30      1Q4.815098^  7OQQQ 

10.301030^2     lO-^'^'^'  =    ()    10=-^^''»^"=    500    lO^-^^'o^o^    200000 

lO'-l^^^l^g         1Q.903090    ^       8       102.954243^       QQQ      1  O^-^O^O^O  =  8000000 

10.602060^4     io'.3oio3o^20     lO^'^o^ceo  ^  4000    10=-^'8'5i=    600000 
The   integral   part  of  the  logarithm  is  called   the  characteristic, 
and  in  case  the  number  whose   logarithm  is  in  question  is  with- 
out decimals, 

(a)  The  characteristic  is  always  less  hy  one  tlian  the  number 
of  figures  composing  it.  But  if  the  number  has  a  decimal  con- 
nected with  it,   then, 


L  0  G  A  R  I  T  K  JI  S  .  I'Lo 

(h)  The  cliaracieristic  is  less  hij  one  than  the  number  of  figures 
on  the  left  of  the  decimal. 

Thus,  the  characteristic  of  2.5  is  0 ;  of  25.67  it  is  1 ;  of  477.3 
it  is  2,  &c. 

141.  By  §  68,  Example  4,  we  may  write  the  following  equa- 
tions : 

i-  =  io-'=      .1    A_  =  io-^=:      .0001    i,-  =  10-'=      .0000001 

_L  =  10-2=    .01    -l-  =  10-^=    .00001    —-  =  10-8=   .00000001 
10=  10*  10* 

i^  =  10-«  =  .001    -^  =  10-''=  000001    ^  =  10-'=. 000000001 

Hence, 

(c)  The  characteristic  of  the  logarithm  of  a  decimal  is  a 
negative  number,  and  is  always  greater  hy  one  than  the  number 
of  cyphers  at  the  beginning  of  the  decimal.     Thus, 

The  characteristic  of  .3  is  —1 ;  of  .053  it  is  —2  ;  cf  .00057 

it  is  — 4. 

To  save  space  the  sign  is  usually  written  above  the  figure,  thus, 

3.602060. 

The  decimal  part  of  a  logarithm  is  sometimes  called  the  man- 
tissa^ and  is  always  positive. 

142.  The  equations  of  §  140  might  be  continued,  and  a  com- 
plete table  be  formed,  including  numbers  as  high  as  we  might 
choose  to  go,  but  such  an  arrangement  would  occupy  far  too 
much  space.  We  may  omit  the  hase,  10,  write  the  numbers  in 
a  column,  and  the  logarithms  to  the  right.     Thys, 

13 


14G 


L  0  c;  A  R  I  T  n  u  s 


1        Table  of  Logarithms  from  1  to  100.         ' 

N. 
1 

Lojr.   ! 
0.000000 

26 

Locr.   1 
1.414973! 

N. 
51 

Log. 

N. 
76 

1 

Lou.   J 

1.707570 

1.880814  i 

2 

0.301030 

27 

1.4313G4| 

52 

1.716003 

77 

1.886491  j 

3 

0.477121 

28 

1.447158 

53 

1.724276 

78 

1.892095  ! 

4 

0.G02060 

29 

1.462398 

54 

1.732394 

79 

1.897G27  [ 

1  5 
G 

0.698970 

30 
31 

1.477121 

I  55 
56 

1.740363 

80 
81 

1.903090  i 

0.778151 

1.491362 

1.748188 

1.908485  i 

7 

0.845098 

32 

1.505150 

57 

1.755875 

82 

1.913814  i 

8 

0.903090 

33 

1.518514 

58 

1.763428 

83 

1.919078  ! 

9 

0.954243 

34 

1.531479 

59 

1.770852 

84 

1.924279  I 

10 
11 

1.000000 

35 

36 

1.544068 

60 

61 

1.778151 

85 
86 

1.929419  1 

1.041393 

1.556303 

1.785330 

1.934498  1 

12 

1.079181 

37 

1.568202 

62 

1.792392 

87 

1.939519  i 

il3 

1.113943 

38 

1.579784 

63 

1.799341 

88 

1.944483 

14 

1.146128 

39 

1.591065 

64 

1.806180 

89 

1.949390 

15 
16 

1.176091 

40 
41 

1.602060 j 

65 

QQ 

1.812913 

90 
91 

1.954243 

1.204120 

1.612784 

1.819544 

1.959041 

17 

1.230449 

42 

1.623249 i 

67 

1.826075 

92 

1.963788'  1 

18 

1.255273 

43 

1. 633468 1 

68 

1.832509 

93 

1.968483 

19 

1.278754 

44 

1.643453 

69 

1.838849 

94 

1.973128 

20 
21 

1.301030 
1.322219 

45 

1.653213 

70 
71 

1.845098 

95 

96 

1.977724  ! 

46 

1.662758 

1.851258 

1.982271 

22 

1.342423 

47 

1.672098 

72 

1.857333 

97 

1.986772 

23 

1.361728 

48 

1.681241 

1.863323 

98 

1.991226 

24 

1.380211 

49 

1.690196 

74 

1.869232 

99 

1.995G35 

25 

1.397940 

50 

1.698970 

175 

1.875061 

100 

m 

2.000000 

^.^  EX  A:\rPLES. 

1.  Multiply  2  !)y  28.     Always  take  tlie  following-  rorm. 
Logarithm  of  2  =  0.301030 

I^garitlim  of  28  =  1.447158 

Logarithm  of  56  =  1.748188 

We  add   the  logarithms  of  2  and  28,  and  find  1.748188,  for 
which  Ave  must  look  in  the  table.     We  find  it   opposite  56. 


L  0  G  A  11 1  T  II  M  S  .  147 

2.  Multiply  11  by  8. 

Logarithm  of  11  =  1.0-41393 

Logarithm  of  8  =  0.903090 

Logarithm  of  88  =  1.911483 

3.  Multiply  2.5  by  3.         {Vide  §140  Z>.) 

Logarithm  of  2.5  =  0.397910 

Logarithm  of  3  =  0.477121 

Logarithm  of  7.5  =  0.875061 

4.  Multiply   .4  by   .0023.  ^Vide  §141  c.) 

Logarithm  of  .4  =  T. 602060 

Logarithm  of        .0023  =  3".  361728 
Logarithm  of      .00092  =  4.963788 

5.  Multiply  17  by   .0005. 

Logarithm  of  17  =  1.230449 

Logarithm  of         .0005  =  4.698970 
Logarithm  of        .0085  =  3.929419 

6.  Multiply  3  by  7,    .8  by  12,    .045  by   .02,^.07  by  1.3,   &c. 

1 14.  Before  giving  other  examples  we  will  explain  the  use  of 
the  table  at  the  end  of  the  book,  which  contains  the  logarithms 
of  all  numbers  between  1  and  10000.  The  characteristic  is 
omitted,  as  it  may  be  easily  supplied  by  the  rules  above,  marked 
(a),   (6),   (c). 

(1.)  If  the  number  consists  of  tJiree  Jigures  with  cyphers  pre- 
fixed or  added: 

Find  these  jigures  in  the  column  marked  N.      Opjiosite  the  num. 
her  ill   the  next  column  is  the  mantissa  of  the  logarithm,   to 
which  prefix  the  cliaracteristic,  ly  (jci),  (?>),  or  (c). 

Since  the  two  left-hand  figures  of  the  mantissa  remain  the 
eaine    for    several    consecutive    logarithms,    they   are    printed    but 


148  L  0  G  A  II I  T  n  M  s . 

once,  in  tlie  column  marked  0  at  the  top.  These  figures  Iclong 
to  the  four  figures  in  all  the  columns.     Thus, 

The  logarithm  of  364  is  2.561101.  The  logarithm  of  365  is 
2.562293.     The  logarithm  of  739  =  2.868644. 

Find   the  logarithms  of  201,   453,   510,   G20,   729,   841,  934, 

and  999. 

The  logarithm  of  3.61  is  0.561101.    The  logarithm  of  .00365 

is  3*. 562293.     The  logarithm  of  73900  =  4.868644. 

Find  the  logarithms  of  28.1,  3.65,  .453,  .0267,  .00384, 
765000,  and  320. 

(2.)  If  the  number  consists  of  foia^  figures  with  cyphers  pre- 
fixed as  decimals,  or  annexed  to  make  up  a  ivhole  iimnher : 

.Find  the  first  three  figures  of  the  number  in  the  column  marlccd 

N.      Opposite   to   these  in  the   column  marked  iviih  a    fourth 

figure  at   the  top  are  four  figures  of  the  mantissa,   to  which 

prefix    the    two    left-hand  figures  of  the  first  column^  and.  to 

the  mantissa    thus  comj)leted  prefix   the    characteristic  hy   (a), 

(6),  or  (c).     Thus, 

The  logarithm  of    3171  is  3.501196.     The  logarithm  of  3172 

=  3.501333.     The  logarithm  of  3173  =  3.501470. 

Find   the   logarithms   of  3845,   4443,   4552,   6854,   7921,   and 

9999. 

If  dots  are   observed   in    passing   to   the   column   marked   with 

the  fourth  figure  at  the  top,  then  the  two  figures  of  the  first  column 
must  be  taken  from  the  line  below.     Thus, 

The  logarithm  of  3166  =  3.500511.     The  logarithm  of  5014 
r=  3.700184.     The  logarithm  of  57.59  =  1.760347. 

Find  the  logarithms  of  5564,  537.6,  53.76,  .5376,  and  .001234. 
(3.)  If  the  number  consist  of  more  than  four  figures : 
Find  the  logarithms  of  the  first  four  figures  as  in  (2). 
Take  the  figures  in    the  column    marked   D   at    the    top,  found 
in    tJte   same   line  ivifh    the   yiumhrr,  and   mnlfiph/   ihrm    hy  the 


L  O  G  A  11 1  T  H  I\I  J3 .  149 

remaining  figures  of  tJie  number  given ^  pointing  off  fom  the 
rigid  as  many  figures  as  are  found  in  the  multiplier. 
Add  the  figures  remaining  at  the  left  to  the  right  of  the  loga- 
rithm already  founds  and  the  sum  ivill  be  the  logarithm  of 
the  given  number,  after  2^^^^fixing  the  proper  characteristic  by 
(a),   (6),  or  (c). 

1.  Find  the  logarithm  of  246891. 
First,  the  Warithm  of  246800  =  5.392345. 

The  number  in  the  column  marked  D  on  the  line  of  the 
number  2468  is  176,  which  multiplied  by  91  gives  160.16. 
We  now  add  the  integral  part  of  the  product  to  the  right-hand 
figures  of  the  mantissa  already  found.     Thus, 

Loirarithm  of      246800  =  5.392345 
176  X  .91  =  160 


Which  gives    Logarithm  of      246891  =  5.392505 

2.  In  the  same  way  find  the  logarithm  of  6789532, 
Logarithm  of    6789000  =  6.831806 
64  X  .532  =  .34 


Logarithm  of    6789532  =  6.831840 

3.  Find  the  logarithm  of  12.347. 

Logarithm  of      12.340  =  1.091315 
351  X  .7  =  216 


Lonrarithm  of      12.347  =  1.091561 
We  add  246   because  245.7  is  nearer  246   than  245.     So  al- 
ways when  the  first  figure  in  the  decimal  is  greater  than  5. 

4.  Verify  the  following. 

Logarithm  of  67895  =  4.831838 
Logarithm  of  68707  =  4.837001 
Logarithm  of  47.306  =  1.674916 
Logarithm  of  432.156  =  2.635640 


150  LOGARITHMS. 

Logarithm  of  . 000432 15G  =  4.635640 

Logarithm  of  78.9102  =  1.897133 

Logarithm  of  4.32195  =  0.635679 

Logarithm  of  .015364  = 

Logarithm  of  123456  = 

Logarithm  of  .023967  = 

Logarithm  of  .111122  =^ 

Logarithm  of  .999999  = 

145.  To  find  the  number  corresponding  to  a  given  logarithm: 
By  (a)  and  (b)  if  the  characteristic  be  0  or  a  j'^ositive  number; 
The  number  of  figures  on   the  left  of  tlie  decimal  in  the   re- 
quired  number  must   be  one  greater  than  is  indicated  J)y  the  char- 
acteristic. 

By  (c).  If  the  characteristic  be  negative,  the  required  number 
is  a  decimal  fraction,  having  the  number  of  cyphers  between  the 
decimal  point  and  the  first  significant  figure  less  hy  one  than  is 
indicated  hy  the  characteristic  of  the  given  logarithm, 

146.  (1.)  If  the  mantissa  of  the  logarithm  can  be  exactly 
found : 

Find   the  mantissa   in   ths  'tahle  and  take  out   the  corresjwnding 
number.     Point  off  as  directed  in  §  145. 

1.  Find  tlie  number  corresponding  to  logarithm  2.928396. 

Ans.  848, 

2.  Find  tlie  number  corresponding  to  logarithm  3.928396. 

uins.  8480, 

3.  Find  the  number   corresponding  to  logarithm  1.928396. 

Ans.    .848. 

4.  Find  the  number  corresponding  to  logaritlim  3.962G06. 

Ans.  9175. 

5.  Find  the  number  corresponding  to  logarithm  3.970114. 

Ans.    .000335. 


LOGARITHMS.  151 

(2.)  If  the  mantissa  of  the  logarithm  cannot  be  exactly  found: 
Take  from  the  table  the  next  less  logarithm  and  the  number  cor- 

responding  to  it. 
Subtract  this  next  less  logarithm  from  the  given   logarithm^  and 

divide  the  remainder  hj  the  number  in  the  column  marked.  D, 

found  VI  the  same  line. 
Annex  the  quotient   to  the  number  alreadtj  taken  out,  and  point 

off  as  directed  in  §  145. 

1.  Find  the  number  whose  logarithm  is  3.123456. 
Form  of  the  Operation. 
Logarithm  of  1328.78  =  3.123456 
Logarithm  of  1328         =  3.123198 


328)258(.TS 
The  next  less  logarithm  tabulated  is  3.123198,  and  the  num- 
ber corresponding  is  1328,  which  we  write  opposite  3.123198. 
Next  subtract  the  logarithms,  and  we  have  258,  which  we  di- 
vide by  328,  found  in  the  column  D.  Annex  the  quotient  .78 
to  1328,  and  it  is  the  required  number,  viz:  1328.78. 
2.  Find  the  number  whose  logarithm  is  1 .  894325. 

Operation. 
Logarithm  of  78.40164  =  1.894325  =  given  logarithm. 

1.894316  =  next  less  logarithm. 


No.  in  column  D  =  55)9000(.164  =  quotient. 

Why  is  the  last  figure  in  the  quotient  4  and  not  3  ? 

3.  Find  the  number  whose  logarithm  is  1.910360. 

Operation, 

Logarithm  of  .813504  =T.  910360 

pro  (last  two  figures  of  next 

"i_     less  logarithm. 

No.  in  column  D  =  53)200(.04  =  quotient. 

4,  Find   tlic  number  VN'hoire  logarithm  is  2.750360. 


152         MULTIPLICATION     BY     LOGARITHMS. 

Ojieration. 

Logarithm  of  .05628078  =Y. 750360  =  given  logarithm. 

54  =  next  less  logarithm. 
No.  in  column  D         77)  6000  (.078  =  quotient. 

5.  Find  the  number  whose  logarithm  is  4.700446. 

Am.   .0005017023. 

6.  Find  the  number  whose  logarithm  is  2.698971. 

Am.  500.0011. 

7.  Find  the  number  whose  logarithm  is  3.602061. 

Ans.  4000.009. 

8.  Find  the  number  whose  logarithm  is  2.650020. 

Ans.  446.704. 

MULTIPLICATION    BY    LOGAKITHMS. 

141^. 

1.  Multiply  24.6  by  25.3.         {Vide  §13^.) 

Operation. 
Logarithm  of  24.6  =  1.390935 

Logarithm  of  25.3  =  1.403121 


Logarithm  of     622.38  =  2.794056 

2.  Multiply  52.74  by  27. 

Operation. 

Logarithm  of       52.74  =  1.722140 

Locarithm  of  27  =  1.431364 


Logarithm  of   1423.98  =  3.153504 

3.   What  is  the  product  of  12  X  34.12  x  .0056  x  5.671  X 
.8123  X  .004  X  23.461 


DIVISION      BY      LOGARITHMS.  153 

Operation, 
Logarithm  of  12  =  1.079181 

Logarithm  of  3^.12  =  1.533009 
Logarithm  of  .0056  =  3.748188 
Logarithm  of  5.671  =  0.753660 
Logarithm  of  .8123  =  F.  909716 
Logarithm  of  .001  ="3.602060 

Logarithm  of  23.46  =  T. 370328 
Log'm  of     .009911568  ="3.996142 

4.  Multiply  23.14  by  5.062.  Ans.  117.1347. 

5.  Multioly  2.581926  by  3.457291.  Ans.  8.92648. 

diyisioj^    by   logaeith:ms. 

14S. 

1.  Divide  24163  by  4567.     (Vide  §13S.) 

Ojjeration. 
Logarithm  of       24163  =  4.383151 
Logarithm  of  4567  =  3.659631 

Loj^arithm  of   5.29078  =  0.723520 

2.  Divide  2  by  3456. 

Operation. 

Logarithm  of  2  =  0.301030 

Logarithm  of         3456  =  3.538574 

Ans.         .000578704  =  T.  762456 

3.  Divide  1   by  256. 

02)eration. 

Logarithm  of  1  =  0.000000 

Logarithm  of  256  =  2.408240 

Ans.  .00390625  ="3.591760 

4.  What  is  the  value  of  .8697  -j-  98.65?  Ans.    .008816., 

5.  Divide  20.76  by  6254.  Ans.    .00476. 


154  LOGAKITHMS. 

ARITHMETICAL     COMPLEMENT. 

149. 
The  arithmetical   complement  of  a  logarithm  is  the  difference 

between  ten  and  the  logarithm.     Thus, 

The  arithmetical  complement  of  3.6020G0   is   10  —  3.602060 
=  6.3979-10. 

The  arithmetical   complement  of  2^698970  is  11.301030;   of 
4^477121  it  is  13.522879. 

15©.  To  find  the  arithmetic;^!  comj^lement  of  a  logarithm: 
Take    the    left-hand  figure  from    9,  and   iwoceed  towards   the 
right,  taJdng  each  figure  from  9  till  the  last  signifcant  figure 
is  reached,  which  must  b^t  taken  from  10. 
Let  X  =   any  logarithm, 
and  1/  =   any  other  logarithm  less  than  x, 
and  c  =   the  arithmetical  complement  of  v. 

By  definition   above   10  — ?/  =  c   or  —  ?/  =  c  — 10. 
Therefore  x  —  7/  =  x  -{-  c  —  10. 
From  which  we  see  that, 

The  difference  between  two  logarithms  is  found  by  adding  to  tlie 
first  logarithm  the  arithmetical  complement  of  the  other,  and 
diminishing  the  sum  by  10. 

1.  Divide  24163  by  4567. 

Operation. 

Logarithm  of       24163  =  4.383151 

Arithmetical  comp  \  ^.^^  _  6.340369         {Vide  §14S,  ex.  1.) 
01  logarithm  ot  j  .     ^  ^  ^ 

Am.         5.29078  =  0.723520  =  sum  by  rejecting  10. 

2.  Divide  .7438  by  12.9476. 

Operation. 
Logarithm  of       .7438  =  T.  871456 

Logarithm  of  12.9476  =  8.887811  arithmetical  complement. 
An^.        0.057447  —  2". 759267  --  sum  by  rejecting  10. 


INVOLUTION      BY     LOGARITHMS.  155 

.      ,         ,         ,  48  X  .75  X  72  X  .0625  „ 


.027  X  120 

Operation. 

Logarithm  of 

48 

=     1.681241 

Logarithm  of 

.75 

=   T.  875061 

Logarithm  of 

72 

=     1.857332 

Logarithm  of 

.0625 

=    2'.795880 

Logarithm  of 

.027 

arith. 

comp. 

=  11.568636 

Logarithm  of 

120 

arith. 

com  p. 

=    7.920819 

Ans. 

50 

=     1.698969=5 

sum  after  re- 
jecting 2  tens. 

6.832     .00634      3642  .657 

4.  Find  the  values  of  -^3^,   ^^aS  '    2O8'  "^^^  70793 

iKA'OLUTION  BY  LOGARITHMS 
151. 

1.  What  is  the  square  of  2.5?  (^Vide  §139.) 

Operation, 
Lofrarithm  of  2.5  =  0.397940 

o 

9 


4?is.  6.25  =  0.795880 

2.  What  is  the  cube  of  32.16? 

Logarithm  of       32.16  =  1.507316 

3 

Ans.  33261.9  =  4.521948 

3.  Find  the  square  of  6.05987.  Ans,  36.72203. 

4.  I^ind  the  ^th  power  of  2.97643.  Ans.  233.6031. 

5.  Find  the  1th  power  of  1.09684.  Ans.  1.909864. 


156  LOGARITHMS. 

EXTHACTIOIC     or     KOOTS     BY     LOGAEITIIMS. 
152. 

1.  Find  the  square  root  of  256.         (^Vide  §139.) 

Ojjeration 
Logarithm  of  256  =  2.408240 

Logarithm  of  16  =  1.204120  =  J  the  logarithm  of  256. 

2.  Find  the  square  root  of  2. 

OjJeration, 
Logarithm  of  2  =  0.301030 

Logarithm  of  1.41421  =  0.150515  =  ^  the  logarithm  of  2. 

3.  Find  the  cube  root  of  2. 

Oj)eratwu 
Logarithm  of  2  =  0.301030 

Logarithm  of    1.2599  =  0.100343  =  ^  the  logarithm  of  2. 

4.  Find  the  4^/i  root  of  2. 

Operation. 

Logarithm  oi  2  =  0.301030 

Logarithm  of    1.1892  =  0.075257.J  =  I  the  logarithm  of  2. 

5.  Find  the  Uh  root  of  7.0825. 

Opei^ation. 
Logarithm  of    7.0825  =  0.850187 
Logarithm  of  1.47923  =  0.170037  =  \  the  logar'm  of  7.0825. 

6.  Find  the  cube  root  of  .023. 

Operation. 
Logarithm  of         .023  ="2.361728 
="3+1.361728 
Logarithm  of     .28438  =  T. 453909  =  -J  the  logarithm  of  .023. 

Here  since  the  characteristic  2  is  negative,  and  the  mantissa 
.361728  positive,  we  cannot"  divide  by  3  as  it  stands.  The  char- 
acteristic must  be  so  modified  as  to  be  exactly  divisible  by  3. 
Kow  2  =  3  -{-  1,  and  we  may  write  the  logarithm  thus,  3  -f-  1 
.361728,  which   i.-^  divisible  by  3. 


EVOLUTION     BY     LOGARITHMS.  157 

7.  Find  the  ^oth  root  of  .0621. 

Operation. 

Logarithm  of       .0621  =^.793092 

=  "5  +  3.793092 

=  10  +  8.793092 

=  15  +  13.793092 

Logarithm  of  .573612  =T. 758618  =  \  the  logarithm  of  .0621. 

Here  '2="5  +  3=To  +  8=15  +  13,  &c.,  either  of  which 
is  exactly  divisible  by  5,  and  gives  the  quotient  1.758618. 

8.  AVhat  is  the  25^/z  root  of  2531000000?        Aiis.  2.37756. 

9.  Find  the  value  of  2}^ 

Operation, 

Logarithm  of  2  =  0.301030 

16 

17)4.816480 
Locrarithm  of  1.92009  =  0.283322  =  \%  of  the  Warithm  of  2. 

10.  Find  the  lOO/A  root  of  5.  Ans.  1.0162. 

11.  Find  the  cube  root  of  2.987635.  Ans.  1.440265. 

(0-!      \     5 
-— jl  Ans.   .146895. 

(119  \  ? 
■y^Y'  -^"«-  1936444. 

14.  Find  the  value  of  \  X  (|)^X  .012  x  (fj)^.  Ans.  .0011657- 

iX  aO^X  .03  X  (151)^ 


15.  Find  the  value  of 


7|x(m)^X  .19  X  (171)^ 

Ans.   .300916. 


153.  Since  the  method  originally  pursued  in  calculating  the 
mantissa  of  logarithms  is  easily  understood,  ■\ve  will  insert  an 
exposition  of  it.     If  we  WTite  the  two  series, 

Is/, 
2??r?, 


0 

1 

2 

3 

4 

5 

&c. 

1 

10 

100 

1000 

10000 

100000 

&c. 

158        EVOLUTION  BY  LOGARITHMS. 

it  is  at  once  seen  that  logarithms  are  a  series  of  numbers  in  arith- 
metical progression  corresponding  to  a  series  in  geometrical  pro- 
gression.    To  compute  the  logarithm  of  any  intermediate  number, 

Find  the  geometrical  mean  of  the  two  terms  of  the  secojid  series 
between  ivhich  the  given  num,ber  is  found. 

Find  the  arithmetical  mean  of  the  two  corresp>onding  terms  of 
the  first  series. 

Again,  Find  the  geometrical  mean  between  this  new  term  and  the 
term  nearest  the  given  number. 

Find  the  corresponding  arithmetical  7nean  in  the  first  series. 
Continue  this  operation  till  the  given  number  becomes  the  geometrical 
mean,  when  the  corresponding  arithmetical  mean  will  be  the  re- 
quired logarithm. 

EXAMPLE. 

Suppose  it  be  required  to  find  the  logarithm  of  9. 
The  geometrical  mean  between  10  and  1  is  l/lO  x  1 

==  1/10"=    3.1G22777 

.   1  _f  0 
The  arithmetical  mean  between  0  and  1  is =rr  A  =  .5 

2 

Therefore  the  logarithm  of  3.1622777  =  .5. 

Again,  The  next  geometrical  mean  is  1/8. 1622777  X  10  =    5.6234132 

1  +  -5 

The  arithmetical  mean  between  1  and  .5  is '—  =  .75 

2 

Therefore  the  logarithm  of  5.6234132  =  .75. 

Sdli/.  The  next  geometrical  mean  is  l/lO  x  5.6235132  ==    7.4989422 

1  +  .75 

The  arithmetical  mean  is  =  .875 

2 

Therefore  the  logarithm  of  7.4989422  is  .875. 

4thli/.  The  next  geometrical  mean  is  l/ro"xT4989422=    8.6596431 

The  arithmetical  mean  is  —^ —  =  .9375 

Therefore  the  logaritlim  of  8.6586431  is  .9375. 


TREATMENT     OF     RADICAL  S.  159 


bthhj.  The  next  geometrical  mean  is  V  10  X  8.6596431  =    9.3057204 

1.9375 

The  arithmetical  mean  is =  .96875 

2 

Therefore  the  logarithm  of  9.3057209  is  .96875. 

Qtlil)/.  The  next  geometrical  mean  is 


1/8.6596431  x  9.3057204  =    8.9768713 

.-,       .    1              •     .9375 +  .96875  n-mor 

The  arithmetical    mean  is   =        .yoolzo 

2 
Therefore  the  logarithm  of  8.9768713  is  .953125. 

Proceeding  in  this  manner,  after  25  extractions,  ■v^'e  should  find 
that  the  logarithm  of  8.9999998  is  .9542425,  and  that  is  the 
logarithm  of  9,  near  enough  for  ail  practical  purposes. 

In  this  manner  Mr.  Henry  Briggs  found  the  logarithms  of  all  the 
prime  numbers  from  1  to  20,000,  and  from  90,000  to  101,000, 
carrying  the  decimal  part  to  14  places.  The  student  will  be  glad 
to  learn  that  in  the  light  of  modern  analysis  all  this  labor  would 
be  lost. 

EVOLUTION  AND  TREATMENT  OF  RADICALS. 
154.  1.  Evolution    investigates    the  metlwd  of  finding  any  root 
of  a  quantiti/. 

2.  A  surd  is  a  quantity  whicJi  requires  a  radical  sign,  or  index, 
to  exactly  express  it. 

3.  A  rational   quantity   requires  no   radiccd   sign   to   exj>ress   it. 

Thus, 

3  is  a  rational  quantity,  but  l/3  is  a  surd, 

X  is  rational,  but  f'  x  is  a  surd. 

4.  The  coefficient  of  a.  surd  is  the  quantify  prefixed  to  it.     Thus, 

5x^,  where  5   is   the   coefficient  of  .t~  or  v  ^. 

5.  A  rational  quantity  may  have   the  form  of  a  surd.      Thus, 

2  =  t/4  =  4i 


IGO  TREATMENT     OF    RADICALS. 

6.  A  surd  Is  in  its  simplest  form  vrlien,  from  the  nature  of  the 
root  required,  the  part  under  the  radical  sign,  or  fractional  index,  is 
the  smallest  possible  whole  number. 

CASE    I. 

155.  To  place  a  surd  involving  an  integral  number  in  its  sim- 
plest form. 

Separate  the  nimiher  into  two  /actors,  such  that  the  root  of 
one  of  them  may  he  exactly  talcen.  Take  this  root  for  a 
coejjicient  of  the  other  factor  affected  hy  the  proper  sign, 

1.  Find  the  simplest  form  of  l/8. 

v/8"=  l/4  X  2  =  V^y.  V^2"=  2V2.     Ans. 

2.  Find  the  simplest  form  of  i^  IG. 

if  16  =  f  8"^  =  #^8  X  1?'2  =  2  #"2;     Ans. 

3.  Simplify  1/T8,  1/32,  V^'SO,  VtI,  Vm,  VWz,  1/2OO,  l/242, 
and  t/288.  Ans.  3  V^,  4  V%  5  V2,  6  1/2,  8  V%  9  V^,  &c. 

4.  Simplify  l/l2,  1/27,  l/48,  j/tS,  i/108,  t/147,  and  l/l92. 

^;;s.  2  1/3",  3  i/'3,  4  V%  &c. 

5.  Simplify  1/20,  V''28,  Vu,  V  117, 1/68. 

^Tis.  2  1/5,  2  1/7,  2  i/n,  3  1/13,  and  2  i/Vf. 

6.  Simplify  i/TG,  "/M,  t/2048,  i/84,  l/l89,  j/iSO,  and  t/338. 

7.  Simplify  t/392,  t/675,  t/1280,  i/2023,  i/3564,  and  l/4693. 

8.  Simplify  1^54,  ^128,  1^250,  1^432,  #"(386,  and  if  l024. 

A?is.  3  f2,  4  r 2,  5  1^2,  6  #^2,  7 1^2;  and  8  1^2. 

9.  Simplify  1^81,  #"135,  1^189,  1^ 29f,  and  1^351 

10.  Simplify  1^320,  ^448,  #"704,  1^'lT25,  and  1^2376. 

11.  What  is  the  square  root  of  8  ? 

Ans.  21/2  =  2  X  1.4H21  =  2.82842. 


TREATMENT     OF     RADICALS.  161 

12.  What  is  the  square  root  of  18  ? 

Ans.  3i/2  =  3  X  1.41421  =  4.24463. 

13.  What  is  the  square  root  of  32,  50,  72,  128,  162,  and  200? 

14.  Find   the   numerical   value  of    all    the    preceding   problems 
by  the  tables. 

CASE   II. 

156.  To  place  a  surd   involving  a  vulgar   fraction  in  its  sim- 
plest form. 

Multiple/  tlie   numerator  and  denominator  of  the  fraction   hy 
such  a  nuraher  as   tcill  render    the    denominator   a  j)erfect 
squai'e,  cuhe,  &c.  as  the  case  may  require. 
Simplify  the  numerator   by  Case  i.,  and  write  the  required  root 
of  the  denominator  under  the  coefficient. 

1.  Find  the  simplest  form  of  V ^^' 

VJ^  =  l/i^  =  t/^^  X  l/3  =  1 1/3 

2.  Find  the  simplest  form  of  t/^. 

l/|  =  l/^  =  Vjx^  =  l/T  X  l/6  =  1 1/6 

3.  Simplify  VI  Vh  y%  Vh  V~h^  &c. 

Ans.  1t/3,  ii/5,  IV\  4v"7,  J,t/11,  &c 

4.  Simplify  V%VJ^,  V~ff^,  l/||,  VVi,  and  l/|l. 

Ans.J^  V%  J-  VG,  ^  V\  -h  V\^~^V\  and  -^-^V^. 
■    5.  Simplify  Vh  ^'h  -^'h  ^h  ^h  and  V^. 

Ans.  1 1/2;  ^  ifi",  J-  f9,  -\  f/%  4  r49,  and  l  fS. 

6.  Simplify  V%  Vh  Vll  -/il,  V'i,  iZ/y,  VJ-,,  and  VI 
'Ans.  il-To,  |t/14,  |i/3,  Jl/22,  ^Vb,  f^V^Z,  /.t/B,  and  \V^. 

7.  Simplify  9  l/p,  6  l/f|,  5  Vj-^,  10 1/^,  and  7 1/^. 

Jns.  4  1^3",  5  V^,  i  VlO,  V^,  and  |  l/2i. 

8.  What  is  the  square  root  of  |  ? 

Ans.  l/r  =  ^  1  '2  =  Jl  of  1.41421  =  .7071. 
11 


162  T  K  E  A  T  M  E  N  T     0  T     R  A  D  I  C  A  L  S. 

9.  What  is  the  square  root  of  ^  ?  Ans.  J  1/2  =  .35355, 

10.  What  is  the  cube  root  of  4  ? 

Ans.  fl=  A#r=  lof  1.5874  =  .7937. 

11.  What  is  the  yalue  of  9  v'p  ? 

Ans.  4l/8  =  4  x  1.73204  =  G.0281G. 

12.  What  is  the  value  of  VJ^  ?  Ans.  -g^  Vdd  =  .20107. 

13.  What  is  the  value  of  l/|f  ?  Ans.  f^Vl  =  .15118. 

14.  What  is  the  value  of  V'X^  ?  J.ns.  %V'b  =  4.02492. 


CASE  iir. 
15^.  To  find  the  root  of  a  positive  algebraic  monomial. 
Take  the' required  root  of  the  coefficient,  and  divide  the   expo 
nent  of  each  letter  by  the  index  of  the  root. 

1.   Find  the  square  root  of  49a*y*'.  Ans.  1x-y^ 


2.  What  is  the  value  of  T/289xy^  Ans.   llxi/. 

3.  AVhat  are  the  values  of  VMl^,  V  44L^»,  and  •l/2MxV  ? 

4.  What  are  the  values  of  t/324^^  1/400^*,  and  1/484^^2  ? 

5.  What  are  the  values  of  l^8^^  l/64xy^  and  i/'lGxY  ? 

6.  What  are  the  values  of  t/243xy,  Vl024:xV,  and  ^^729:^^' 

CASE   IV. 

15S.  To  find  the  root  of  a  negative  algebraic  monomial. 

(1.)  ;}-  a  X   -|-  «=  +  a^,  and  —  «  X   —  a  =  -\-  «*,  .-. 

TJie  even  root  of  a  negative  quantity  is  impossible. 

(2.)   —  a  X  —  «  X  —  a  =  —  a^  .-.  1^—  a^  —  —  a.     Hence, 

The  odd  root  of  a  negative  quantity  is  negative. 

(3.)  ±  a  X  dca  =  a^  .-.  l/a^  =  d=  a.     Hence, 

TJie  even  root  of  a  j^ositive  quantity  is  positive  or  negative. 

When  the  double  sign  ±  occurs  two  or  more  times  in  the  same 
equation,  the  upper  signs  must  not  be  confounded  with  the  lower. 
Thus,  db  «  qp  Z>  =  it  c  means  +  a  —  b  =  c,  ov  —  a  -\-  b  =  —  c. 

1.  Find  the  cube  root  of  —  Sx^^y-*.  Ans.  — 2x*y^. 


T  R  E  A  T  M  E  N  T     0  T     11  A  D  I  C  A  L  S.  163 


2.  What  is  the  value  of  ^/— 32xy°?  Ans.   ^2x7/\ 

3.  What  are  the  values  of  t/lQxy,  f—  'Zlx^f,  and  l/l69xy  ? 

4.  What  are  the  values  of  '^'^V';  V^^^\  and  i/Qo^^^  ? 

6.  W^hat  are  the  values  of  J/_16a*,  ^— 2187x'i*j/^S  and  l/529^? 
6.  What  are  the  values  of  V^^,  T^6i^V^^  and  T^— 216xy2  ? 

CASE   V. 

159.  To  simplify  algebraic  monomials  whose  root  cannot  be  ex- 
actly taken. 

Swiplifi/  the  numerical  part  l>y  I.  or  II. 

Divide  each  exponent  by  the  index  of  the  root  to  be  taken,  and 
write  the  letters  with  the  quotient  for  an  exponent  on  the  outside 
of  the  radical  sign,  and  the  letters  with  the  remainder  for  an 
exponent  under  the  radical  sign. 

EXAMPLES. 

1.  AVhat  is  the  simplest  form  of  j/lSxy?  Ans.  3xyi/2. 

2.  What  is  the  simplest  form  of  l^54xy  ?  Ans.  Zxy  1^2p. 
8.  Simplify  {/Z2^,  ^192xy,  l/py,  and  if py. 

4.  Simplify  l/py,  V j\x*y\  V^^^xY,  and  flQxY- 

5.  Simplify  lfl28x^y,  ^^OSy^,  T^py,  and  /py. 

6.  Simplify  i/py^  l/py^  5l/py,  and  15l/8^. 

7.  Simplify  l/44xy,  l/75xy,  l/Sxy^  and  7  l/28xy. 

8.  Simplify  l/50xy,  l/200xy,  |/243^,  and  l/|^p: 

9.  Simplify  l/Jxy,  l/fxy,  l/fa/^;  and  Vxyz"^. 

CASE  Yl. 

160.  To  add  radical  quantities. 
Simplifi/  each  expression  hy  Case  V. 

If  the  radical  parts  are  then  the  same,  add  the  coejicients  and 
prefix  the  stun  to  the  common  radical. 


164  TKEATMENT    OF    RADICALS. 

If  the   radical    parts   are  not  the  same,  unite  the  quantities  hy 
the  proper  sign. 

EXAMPLES. 

1.  Add  V^TI^\  t/48^^  VI'^xY  and  Vl^'^xy. 

Operation. 


l/75^^    =    bx't/  VS 
\/ld2xY  =    Sx'^f/V^ 

20x^1/  1/3  =  the  sum. 
2.  Add  together  #^54^,  and  f^VlSx'y. 

Operation. 

•^54^    =3.ry    f/2f 
lfl28x»y=  4:ry  "if  25/^ 

(3^^+4xy)  if  %^  =  the  sum. 

8.  Add  together  l/320xy,  l/lSOxy,  l'''245xy,  and  l/20xy. 

J.71S.  25:ryi/5. 

4.  Add  together  l/28xy,  -/CS^y,  and  T/ll2xy. 

5.  Add  together  l/99xy,  l/275xy,  l/l76xy,  and  v^Iixy. 

6.  Add  together  V  py,  l^py,  l/IiOxy,  and  y'315xy. 

7.  Add  together  V'py,  V^PV;  V^PV;  and  V'-^^xy. 

8.  Add  together  l/fxy,  l/py,  i  T/84xy,  and  Jl/l89xy. 

9.  Add  together  l/py,  l/py,  T/|gxy,  and  l/20^. 

10.  Add  together  f'Uxy,  if  192^,  1^81^,  and  #'375xy. 

11.  Add  together  ]/32xy,  {/l62xy,  f/512.ty,  and  i/l250^«. 

12.  Add  together  f^py,  J^^V^j  f^S^j  ^^^  f^^^- 


TREATMENT     OF     RADICALS.  165 

13.  Add  together  v'^xY,  VixY,  and  \/Jx^^. 

14.  Add  together  V-\xYp  V^x^tj^,  and  V'^^x^- 

15.  Add  together  fMxY,  #^32xy,  and  f\OxY- 

16.  Add  together  l/20xy,  l/l^^y,  l/86xy,  and  l/iixy. 

17.  Add  together  VlxYz'j  V'lx-fz^  and  t/^^V^'. 

18.  Add  together  l/49xy,  l/6ixy,  V^lxY,  and  v'^^y. 

19.  Add  together  \'^2x^  +  ^.ry  +  2^^.  and  ■l/2x^  —  4:ry  +  2_y'. 

J.71S.  2xl/¥- 

CASE    VII. 

161,  To  subtract  radical  quantities. 

Simjplify  each  expression  hi/  Case  Y. 

If  the  radical  parts  are  then  the  same^  subtract  the  coefficients 
and  prefix  the  difference  to  the  common  radical. 

If  the  radical  parts  are  not  the  same,  express  the  difference  hy 
the  projyer  sign. 

EXAMPLES. 

1.  From  Vl^xY  take  V^xY- 

Operation. 


VlSxy  =  3j7/y2 
V^"'    =  'IxyV^ 

xyy2i  =  the  difference. 

2.  From  l/py  take  l/py.  Ans.  J^V  i/2. 

3.  From  l/63:cy  take  l/28xy.  ^ws.  xY  i/7. 

4.  From  l/SOxy  take  l/20xy.     5.  From  l/fxy  take  V^xY- 
6.  From  l/py  take  l/f^y.        7.  From  l/||xy  take  i/'if^y. 
8.  From  t/3^«  take  l/py.        9.  From  if  27aV  take  if  8^». 

10.  From  #'l92^iy5  ^^ke  f^^i^^. 


166  TREATMENT     OF     H  A  D  I  C  A  L  S. 

11.  From  f/lO^y  take  \/ x^if.      12.  From  i/82xV^  take  V^^y^^- 

13.  From  f/ei^^  take  Vxh/''. 

14.  From  "/j^^^  take  l/j^xy. 

15.  From  l/^^^y  take  l/^^cc*/.  16.  From  2l/|"take  3l/|. 
17  From  V%  take  j/^^. 

18.  From  V'lx'  +  4£cy  +  2^  take  l/2x2  _  4^^  ^  4^2^    ^,^5.  2?/i/2, 

CASE    VIII. 

162.  To   multiply   radical   quantities;    multiply  tlie  coeflicients 
and  also  the  radicals. 

Simjolifi/  the  result  hy  Case  V. 

EXAMPLES. 

1.  Multiply  7  l/5.x-y  by  4  l/fx^. 

Operation. 

7  V^x^if 
41/ 


x-y 


28  l/2xy  =  28xy  1/2  =  product. 
2.  Multiply -I i/pvi^yil/p^ 


Operation. 


3  V-i^xhfz^  =  -Jt?/2;  i/j^%  =  -^-^xyz  l/l5  =  product. 

3.  Multiply  4-^2  by  2l^4.  Ans.  16. 

4.  Multiply  t/.t  by  l/o^.  Ans.  x. 

5.  Multiply  3  V^  by  4  l/20x.  ^7is.  120;r  l^y. 

6.  Multiply  7  t/3^  by  8  l/l6pp  ^«s.  392xj^»  i/^ 

7.  Multiply  5  Vlxy^  by  -|  l/27a^.  ^ws.  ^xy"^  j/y- 

8.  Multiply  5  l/|^  by  y^^  1/40^  ^«s.  2xy\ 


TllEATMENT     OT     11  A  D  I  C  .1  L  S.  IGT 

9.  Multiply  V'"^^^^~+1  by  j/x^-}-  1.  Ans.  a;^  +  1. 

10.  Multiply  l/:r  -f  1  by  l/ic  —  1.  .I'^s.  j/ic^  —  1- 


11.  Multiply  Vx^  +  a-  by  V  x^  — 


12.  Multiply  "l/x  +  l/y  by  l/a:  —  "j/?/. 

13.  Multiply  i/3  +  l/2  by  l/3  —  V  2. 


14.  Multiply  1/7  +  1/24  by  V7  —  VU- 

15.  Multiply  1/5  +  1^2  by  1  '5  +  V2. 

16.  Multiply  V'5.>:  +  V4:]/  by  l^Sx  —  v'J^. 

17.  Multiply  1/7+  1/3  by  V7  —  VS. 

18.  Multiply  1/15  +  l/'2  by  l/l5  +  y/'J. 

19.  Multiply  l/2x  +  V  %  by  t/2x'  +  |/3y. 

20.  Multiply  1/^2  +  vTS  by  l/2  +  y  18. 

21.  Multiply  i/5x  +  VSi/  by  l/5x  +  l/Sy. 

22.  Multiply  4  i/2^  —  3  VT?/  by  5  j/si  +  7  T/l8y. 

23.  Multiply  5  V2x  +  3 1/1%  by  3  i/ISTc  —  5  ;/%. 

24.  Multiply  j/x  +  y'l/  by  l/:£  —  V^. 

2o.  Multiply     .-    —  by      ._ 

Vx  —  7        l/rc  —  4 

26.  Multiply  ^"  +  Syj$--'^ 


l/a:  +  7        l^x  —  7 

27.  Multiply  X  4-  V  ^y  -f-  3'  by  :«  —  V  xi/  -\-  1/.    Ans.  x^-{-  xi/  +y^ 

28.  Multiply  —  a  +  l/a^^+l;  by  —a  —  Vo^+h.         Ans.   —  h. 


29.  Multiply  a  +  y  a^  +  Z>  by  a  —  y  a'^  +  b.  Ans.   —  &. 

30.  Multiply  6  +  l/36  +  2  by  6  —  |/36  —  2.  Ans.  —  2 

31.  Multiply  yx  +  ^  +  1/ x  —  3^  by  I'^x  -\-  y  —  v  x  —  y. 

Ans.  2y. 

32.  Multiply  Vx  +  Z  by  Vx  —  o,  also  Vx  +  2  by  j/^—  2;  and 
l/o;  +  4  by  "l/x  —  4. 


168  TREATMENT    OF     RADICALS. 

CASE  IX. 
163.  To  divide  radical  quantities. 
Simjplift/  each  expression  hy  Case  Y. 

Prefix   the    quotient  of  the    coefficients  to  the   quotient  of  the 
radical  parts,  and,  if  necessary,  again  simplify  for  the  final  result. 

EXAMPLES. 

1.  Divide  i  l/28j:y  by  |  V^^xy\ 

Operation. 
I  l/28xy  =  |:r^2 1/7^ 
I  VM^  =     2yVTx 

g^y  l/y     =  quotient. 

2.  Divide  t/20:^  by  V^^.  Ans.  |  l/IoZ 

3.  Divide  Vx^  -\-  y^xij  +  Vxy^  by  l/^.       J.ws.  i/^  +  |/y  -j-  y. 

4.  Divide  V'^^xY  +  t/63^  +  VU2^  by  j/T^ 

5.  Divide  4  ]/2^-^  by  \  Vpy.  Ans.  2  l/2^. 

6.  Divide  1/I6  +  V  4  by  4  |/J.  ^7is.  |. 

7.  Divide  v''l9  —  l/9~by  2.  Ans.  2. 

8.  Divide  x''  +  o-y  -|-  y^  by  cc  +  V  '-^^  -f  y-     •^•^'^s-  -^  —  vscy  +  y* 

„    ™    ,^,        ,        ,1/54  81/50  121/28  151/378  l/J       ,l/| 

9.  Innd  the  values  01 ——3, ;^:::-, — ^, 7=:-; — ^,and  — Z.. 

VQ    41/2       SV7       bVQ       V\         Vi 

Ans.  3,  10,  8,  9  VY,  1 1/6",  and  |. 

ia   T7-    wi        1         f  ?l/l8  fV"!  -ij/J          1/24-31/1 
10.  Find  the  values  of  ^^ — -^,  ^ — -i,  -- — p^,  and !— = — ^. 

A  1/2       1  1/3'  J.  -,/i'  J  -i/i 

Ans.  4,  1 1/5,  2,  and  10. 


TREATMENT    OF    RADICALS.  169 

CASE   X. 

164.  To  reduce  a  fraction,  whose  denominator  is  a  binomial 
containing  radical  quantities,  to  an  equivalent  fraction  without 
radicals  in  the  denominator. 

Multiply  both  the  niunerator  and  denominator  of  the  fraction 
hy  the  denominator  with  one  sign  changed. 

EXAMPLES. 

3 

1.  Reduce  the  fraction  — = ;=.  to  a  fraction  having  no  radi- 

Vb  +  i/2 

cals  in  the  denominator. 

Operation, 

3  ^i/5-t/2^  31/5-31/2^^- __^^^3^^3^^ 


l/5  + 1/2       1/5  —  1/2  3 

2 
2.  Find  the  value  of 


1/3  +  1/5 

Ans.  ^^^-^^  =  v^  -  V^3-  =  .50401. 
—  2 
3 

3.  Find  the  value  of  ;=.  Ans,  .31866. 

8  +  1/2 

5  8  5 

4.  Find  the  values  of -=,     — = 7=,     7^,     and 

7  _  i/40      1/3  -  1/7      9  -  1/8 

1/3  +  -/2 
1/3  -  1/2' 

;;    -P-   ^  .1,         1  1/3-1/21/5  +  1/3  1/5-1/3 

5.  Find  the  values  of  —= ^,  —~ 7-,  and  -— = 7=. 

1/3  +  1/2  1/5  —  1/3  1/5  +  1/3 

6.  Find  the  values  of  — -= 7^,  and  —^ 


1/5  —  1/2         Vh  —  VS 

Ans.  3.65028,  2.80588. 

n    1?'  A.i.       1         P  1^7  +  1/5       ,  2i/n-3^13 

7.  Find  the  values  of  . 1..  and — — r* 

1/7  _  1/5         21/11 +  3l/l3 

15 


170  T  E  E  A  T  M  E  .N  T     OF     RADICAL  S. 


a \/a^ x'^ 

8.  Given   — • izzzzzzij  ^^  ^^^^  ^^^  denominator  of  radicals 

2a2  —  a:'  —  2al'^a'  —  x« 
-4?is. 

9.  Given  ,  to  tree  the  denominator  oi  radicals 

yx  —  yx  —  a 

,        2x  —  a  -{-  2  Vx^  —  ax 
Ans.  • ■ • 


CASE    XI. 

165.  To  find  the  square  root  of  a  polynomial. 

1.  If  necessary,  arrange  the  polynomial  witli  reference  to  a  given 
letter,  and  place  the  square  root  of  the  first  term  to  the  right  of 
the  polynomial,  for  the  first  term  of  the  root.  Square  this  term, 
and  subtract  it  from  the  polynomial. 

2.  Double  this  first  term  of  the  root,  and  place  it  on  the  left 
of  the  remainder  for  a  part  of  the  divisor;  divide  the  first  term 
of  the  remainder  by  this  double  of  the  root,  and  place  the  quotient 
in  the  root  as  the  second  term,  and  also  at  the  right  of  the 
divisor. 

3.  Multiply  the  wJioIe  divisor  by  the  second  term  of  the  root, 
and  subtract  the  product  from  the  remainder. 

4.  Double  the  whole  of  the  root,  and  write  the  result  to  the 
left  of  the  remainder,  as  a  part  of  the  divisor;  divide  the  first 
term  of  the  remainder  by  the  first  term  of  the  partial  divisor,  and 
place  the  result  as  the  third  term  of  the  root,  and  also  at  the 
right  of  the  partial  divisor. 

5.  Multiply  the  whole  divisor  by  the  third  term  of  the  root, 
and  subtract  the  product  from  the  last  remainder. 

6.  In  a  similar  manner  find  other  term». 


T  K  E  A  T  M  E  N  T     OF    K  A  D  I  C  A  L  S.  171 

EXAMPLES. 

1.  Vfhat  is  the  square  root  of  x^  +  4x3  4.  Gx^  -f  4x  -f  1  ? 

Ojieration. 
x*  +  4x3  -i-  6x2  ^  4.^  ^  X  I  x2  4-  2x  +  1.  J[?is. 

X* 


2x»  +  2x 


4x3  +  6x2  -f  4x  +  1 

4x3  _|_  4^2 


2x»  4-  4x  +  1  I      2x2  4.  4^  _^  1 
I      2x2  4-  4a:  4- 1 

2.  What  is  the  square  root  of  x*  —  2x3  4-  Sx'  —  2x  4-  1  ? 

Operation. 
X*  —  2x3  4-  3x2  —  2x  4-  1  I  ^2  __  ^  ^  1 

X* 


2x2  — X 

—  2x3  4-  3x2  —  2x  4-  1 

—  2x3  4-    x2 

2x2- 

-2x4-1 

2x2  —  2x  4-  1 
2x2  _  2x  4-  1 

3.  What  is  the  square  root  of  a:2  4-  2x'  4-  3x*  4-  2x^  4-  x«  ? 

Ans.  X  (1  4-  a;  4-  ^')' 

4.  What  is  the  square  root  of  x2  —  2x^  4-  3x*  —  2x5  4-  x"'  ? 

Ans.  X  (1  —  X  4-  ^^)« 

5.  AVhat  is  the  square  root  of  x'  4-  4^2  _^  9^2  _^  4^^  ^  q^^  _^ 

12^/^?  Ans.  x  +  2i/  +  Sz. 

6.  What  is  the  square  root  of  x*  4-  3x2  _j_  2x^  _  2x  4-  1  ? 

A7IS.     X2   4-   X  1. 

7.  What  is  the  square  root  of  x2  4-  2xy  4-  ^2  4,  2x  4-  2^  4-  1  ? 

-4.715.  cc  4-  y  +  !• 

8.  What  is    the   square    root  of  x^  —  2x^  —  x*  4-  4ar'  —  x'  ~ 
2a:  4-1?  ^«s.  x3  __  ^3  __  2c  4- 1. 


172  I  M  A  G  I  .\  A  K  Y     Q  U  A  X  T  I  T  I  E  S. 

9.  What  is  the  square  root  of  1  —  x"^  ? 


X^  T"*  QJ^ 

Ans.  1  _  -  -  11  -.  _ ,  &c. 


MISCELLANEOUS. 
166. 

1.  Complete    the    square  of  x-  -f  ^j^.r,  and  take  the  square  root. 


(  Vide  132,  8.)  Ans.  V x-  +  'Zjyx  -\-  p^  —  x  -\-  p. 

2.   Complete  the  square  of  x-  -f  2x,  and  take  the  square  root. 

Ans.  X  -\-  1. 
o.   Complete  the  square  of  :c'  —  3.r^  and  take  the  square  root. 

Ans.  X  —  4. 

4.  Complete  the  square  of  x^ j ^,  and  take  the  square  root. 

An^.  X -' 


2r  —  71V 


CASE    XII. 

1 

IMAGINARY   QUANTITIES. 
167.  An   imaginary  quantity   is    an    indicated    even   root  of  a 
negative  quantity.     Its  general  form  is — 


±  A  V  -  1 

where  A  is  either  rational  or  radical. 

The  rules  for  multiplying  imaginary  quantities  depend  upon  the 
fundamental  principle  that  the  square  root  of  a  quantity  multi- 
plied hy  its  square  root  produces  the  quantify  itself.     Thus, 

From  this  we  have  the  following  table  of  equations. 
— \/— xX     s/—y=--\/z  v/HTx     v^vZ—l  =— v^X— "•=     v^.  4. 


IMAGINARY    QUANTITIES.  173 

—\/^  X—\/-~i=—\/x\/^  X  —  >/z\/^  =     xX  —  1=  —  2;.6. 

\^~z  X— v/^=     \/?v/^l  X  —  y/z\/^l  =  — a;  X  —  1  =      a;  .   7. 

'-\/^X    v/^=— v^^n/^  X      \^y/^  =  —z  X  — 1=      X  .  S. 

Hence,  like  signs,  on  the  outside  of  two  imaginary  quantities, 
produce  minus,  and  unliJce  signs,  p?i<s,  and  the  product  is  not 
imaginary. 

It  must  be  remembered  that  this  apparent  exception  to  the 
common  rule  of  multiplication,  applies  only  when  the  quantities  to 
be  multiplied  are  hoth  imaginary;   for 

|/8  X  /^^o=  /^nX=  v/To  N^^^  and 

/8  X  /^^=  /^^  =  3  /^n",  &c. 

EXAMPLES. 


1.  Multiply  —1-1-  >/—  3  by  —  1  4-  |/—  3. 

Operation. 

-1-1-  /ir3 


1-  y/zr^ 

_  2  —  2  /^^^.  ^ns. 
2.  Multiply  —  2  —  2  /^Ts"  by  —  1  -f-  /^^S 

Operation. 
_  2  —  2  /^^^ 
-1+      /^Ts 


2  4-2  /— 3 
_  2  /^I^  +  6 
8  =  24-6.  Am. 

lb* 


174  IMAGINARY    QUANTITIES. 

By  comparing  these  two  examples  we  see  that  ( —  1  +  •>/ —  3)^  =  8. 

Wc  should  also  find  (—  1  —  V^^oJ  =  8. 

3.  Multiply  1/1^2  -f.    /^ITl   ^    y/'ZT^  by  n/^^T^  —  V^^ 

—  v'z:5. 

operation. 

—  2  —  -1/2    —  i/ro 

+  1  +  1/2      +  i/lo  +  1/6 

+  5  +  t/5 


4  +  2i/5  =  A71S. 

4.  Multiply  5i/7^    +   81/^^28    +   2  i/7I~7  by  61/^^4 
—  3l/^9  +  St/ITt. 

5  i/ZTl  +  3  v/Z:28  +  2  i/^^l  =  5  i/^Tl  +  8  t/^=^  . . .  by  VI. 

61/zn  _  3  i/iTg  +  5  T/:r7  =  3  ]/:rT  +  5  v":^ . .  .by  vii 

-  15  —     24 1/7 

—     25t/7--280 


—  295  —     49  Vl  =  ^ris. 


6.  Multiply  V—  X  +   V—  xi/  +   ;/— ?/  by  V—  x  —  i^—  aj/ 
+  V —  y.  Ans.  —  X  -\-  xy  —  y  —  2  yxy. 

6.  Find  the  value  of  (l/^^  +  V^^y.      Ans.  —  7  —  2  i/ICh 

7.  Find  the  value  of  (l/^^  +  V^^y.   Ans.  —  13  —  4  /la 

8.  Find  the  value  of  (V' —  27  +  1/^^^)^  JLns.  —  48. 

9.  Find  the  value  of  (;/— ~28"  —  V^^^y.  Ans.  —  7. 
10.  Find  the  value  of  l/^HJ  —  •/7r5)l  tIjjs.  —18  +  2  l/65^ 


IMAGINARY    QUANTITIES.  175 

11.  Find  the  value  of  (v''^^^ -\-  V^^)  (V-l-V^).  Arts.  1. 

12.  Find  the  value  of  (V^^  +  V^^)  (l/^^  —  V^^'). 

Arts.  2 

13.  Find  the  value  of  (—  1  —  V^^f. 

Ans.  (—  1  —  V^^^y  =  (—  1)'  —  3  (—  1)^  V^o  +  3  (—  1) 
l/^Ts^  ~  /31;3  =  8. 

14.  Find  the  value  of  (—  1  +  V^^f-  Ans.  8. 

15.  Find  the  different  powers  of  l/ —  1. 

Ans.  V  —  1  X  V  —  1  =  —  1  —2.nd  power  of  V  —  1. 

—  1  X  t/^^^  =  —  1^^^^  =  3rfZ      ^^        "    i/^^nr. 


—  ]/_  1  X  V—  1  =  1  =  4^/i      "       "   t/—  1. 

1  X  v^^  =  i/^^^"!    =  5^A    '^    "  i/^^nr. 

And  since  the  quantity  and  its  5th  power  are  the  same,  it 
follows  that  the  2nd  and  6th  powers  must  be  the  same,  so  also 
must  the  Srd  and  7th,  the  4th  and  8th,  &c.,  be  alike. 


16.  Find  the  value  of  (2  +  5  V—  1)*  +  (2  —  5  v'—  1)*. 

Ans.  82. 


17.  Find  the  value  of  (1  +  V  —  WJ  —  (1  —  v'  —  11)^ 

An^.  =  992. 

IS.  Find  the  value  of 7= 1 7^=^-   ^"s-  2. 

i(l-f/-l)       1(1-  /-I) 


CHAPTER  VII. 

EQUATIONS  OF  THE  SECOND  DEGREE. 

168,  An  equation  of  the  second  degree  is  one  involving  the 
second  power  of  the  unknown  quantity. 

Such  equations  may  be  complete  or  incomp)lcte. 

A  complete  equation  is  one  involving  both  the  first  and  second 
degrees  of  the  unknown  quantity.     Thus,  x^  +  ^^^  ==  ?• 

An  incomplete  equation  is  one  involving  only  the  second  degree 
of  the  unknown  quantity.     Thus,  2ax^  =  q. 

169.  To  find  the  value  of  x  in  an  incomplete  equation. 
Proceed   exactly   as   with   simple   equations   of  one   unlcnown 

quantity^  and    tahe    the    square   root  of  the  final   equation 
for  the  value  of  x. 

EXAMPLES. 

1.  Given  x'  —  192  =  -r,  to  find  the  values  of  x. 

4' 

Operation. 


X 

'  —  192  =  —  (1)  =  given  equation. 


X 

4 


4x'  -  768  =  x»  (2)  =  (1)  X  4. 

Sx'  =  768  (3)  =  (2)  reduced. 

x»  =  256  (4)  =  (3)  -T-  3. 

X   =  d=  16  (5)  =  l/(4).        (  Vide  15S.  3.) 
ne 


EQUATIONS    or    THE    SECOND    DEGREE.       177 


2.  Given 8  = f-lO,  to  find  the  values  of  x. 

3  9 


x"^ 
3.  Given  8  4-  5x'  = \-  ^x""  +  28^  to  find  the  values  of  x. 

5 


Ans.  cc  =  ±  9. 
3  values  of  x. 

Ans.  £c  s=  db  5. 


X  ^, oX 


4.  Given  —  =  —  671  H ,  to  find  x.  Ans.  x  =  dz  7. 

8  ^  ^    2 


/y2  y  /y-«2 

5.  Given  —  4-  4  =  ■ 15i,  to  find  x.  Ans-.  a:  =  db  3. 

5  3  ' 

6.  Given  ?^!_±_5  _  '^'  +  ^^  =  117  -  5x^  to  find  x. 

8  3 

Ans.  cc  =  ±  5. 

7.  Given  ^'  +  12  =  —  +  37|,  to  find  x.       Ans.  x  =  ±z7. 

3  7 


8.  Given  -  -  1  = 

3 

9.  Given  ^   +  ^ 

ic'^  —  7x 

4^2 
= 1-5    to  find  X.               Ans.  x  =  da  S. 

27 

...          trt    Tinn     'y 

^2  ^  7_^       ^2  _  73 

Ans.  ic  =s  ±  9. 

10.  Given  (x^  -f  1)^  =  25,  to  find  x. 

Operation, 
(x^  -f-  1)^  =  25  (1)  =  given  equation, 

x^  +  1     =  ±  5  (2)  =  1/(1). 

x"    =  4  or  —  6  (3)  =  (2)  reduced. 

X    =  ±  2  or  ±  t/^Tg    (4)  =  1/(3). 


p.       (^  +  18)' 
Given  -^ 

28 

= 

4^2 

— ,  to  find 
63' 

Operation. 

a;. 

(x  +  18)2       4^2 
28              63 

(1)  =  given  equation 

{x  -f  18)2  _  4x' 
4               9 

(2)  =  (1)  X  7. 

a-  4-  18     _  ^ 
2 

2x 
3 

(3)  =  v/(2) 

X  =  54 

or 

—  1^ 

(4)  =  (3)  reduced. 

178        EQUATIONS    OF    THE    SECOND    DEGREE. 


12.  Given — -  =  — ,  to  find  x.  Ans.  cc  =  5  or  —  i|. 

80  1 6S  ? 

13.  Given =  5,  to  find  x.  Ans.  cc  =  15  or  4i.f . 

5(a;  —  7)2         3^2  ^' 


^,     ^.  rz;2  -f  3  2x2  —  13        ^    , 

14.  Given  =  ,  to  find  x.         Ans.  x  =  ±  4. 

2x2  —  13  a;2  4-  3 

15.  Given  ax^  =  h,  to  find  x.             Ans  x  =  =fc  ^/-  =  -  ■l/a6. 

16.  Given  x^  -f  ai>  =  7x2,  ^q  gj^^j  ^  ^^^^    ^  =  |  T^6a6. 

17.  Given  (x  -f  a)2  =  2ax  +  h,  to  find  x.  Ans.  x  =  ±  i/fe—a^. 

18.  Given  ^     ^    ^   =  c\  to  find  x. 

(x2  —  by 


Ans.  X  =  =fc  J'L±^  or  v/^^  -  ^. 
\c  -  1         N/c  4-  1 


+ 

in    n-         5c  +  a        X  —  a  10«2 

19.  Given H =  ,  to  find  x. 

X— a        x4-<^        ^^  —  <^^ 

^?is.  X  =  dz  2a. 

rtrt    y-..         •'^  —  ^        ^  —  2x        x2  -f  Jx  „    - 

20.  Given =  ,  to  find  x. 

a  X  —  a  x}  —  a? 

Ans.  X  =  ±  y/ ah. 


PROBLEMS    PRODUCING    INCOMPLETE    EQUATIONS    OF   THE 
SECOND    DEGREE. 

I'yO.  1.  Find  a  number  whose  |  multiplied  by  its  |  will  be 
equal  to  2520. 

Let  X  =  the  number:  then  --    x  —  =  =  2520. 

6         7  42 

#  .-.  X  =  ±  84. 

2.  Two  numbers  are  to  each  other  as  3  to  7,  and  the  sum  of 
their  squares  is  522.  What  are  the  numbers?  Let  3x  and  7x  = 
the  numbers. 

The  equation  is  9x2  ^  i^(^^.  ^  522. 

.-.  X  =  3,  and  8.r  =  9,  and  7x  =  21.      .4ns.  9  and  21. 


EQUATIONS     OF    THE    SECOND    DEGREE.       179 

8.  Two  numbers  are  to  eacli  other  as  |  to  |,  and  the  differ- 
ence of  their  squares  is  153.     What  are  the  numbers  ? 

Ans.  24  and  27. 

4.  If  4  be  added  to  a  certain  numbei^and  also  subtracted,,  the 
product  of  the  sum  and  difference  will  be  609.  "What  is  the 
number?  Ans.  25. 

5.  If  9  be  added  to  a  certain  number  and  also  subtracted,  | 
of  the  product  of  the  sum  and  difference  will  be  162.  What  is 
the  number?  Ans.  18. 

6.  A  and  B  start  from  different  points  at  the  same  time,  and 
travel  towards  each  other.  On  meeting,  A  has  travelled  20  miles 
farther  than  B,  and  A  would  have  gone  B's  distance  in  75  hours, 
but  B  would  have  travelled  A's  distance  in  108  hours.  What 
distance  had  been  travelled  by  each  ? 

Let  X  =  B's  distance;  then 
cc  -f  20  =  A's  distance  .-. 

—  =  A's  progress  per  hour,  and 

^±^  =  B's         "  ''        " 

108 

Now  A's    distance  :  B's   distance  : :  A's   hourly  progress  :  B's 

hourly  progress;  or 

a:       X  -f  20 

75         108 

(x_+_20y_  0^ 

108        ~  75' 

Ans.  A  120,  B  100  miles. 

7.  Two  numbers  are  to  each  other  as  a  is  to  h,  and  the  sura 
of  their  squares  is  c.     What  are  the  numbers  ? 

The  equation  is  a^x^  -f  h^x-  =  c,  whence 

X  =     ^  ;  then  the  numbers  are  —-===.  and  —-==^ 


180       EQUATIONS    OF    THE    SECOND    DEGREE. 

8.  Two   numbers   are   to   each   other   as   a   to   h,  and  the   dif- 
ference of  their  squares  is  c.     What  are  the  numbers  ? 

.  aye  ,        ^l/c" 

Ans.  :  and 


9.  A  man  drew  from  a  cask  of  wine  containing  a  gallons  a 
certain  quantity^  and  then  filled  it  with  water.  He  then  drew 
of  the  mixture  the  same  number  of  gallons  as  before,  and  again 
filled  the  cask  with  water.  Having  done  the  same  a  third  and 
fourth  time,  he  has  h  gallons  of  wine  left  in  the  cask.  How 
many  gallons  of  wine  were  drawn  off  each  time  ? 

3J  1211  Ii2j.  131  1 

Ans.   a4  (a4  _54)^a4  h"^  (a4  —Z>4)^cj4  ^4  (^<^4  _  2,4)^  54  (^^4  __  2,4). 

If  a  =  256  and  h  =  81,  then  64,  48,  ZQ,  and  27  are  the 
answers. 

If  a  =  625  and  h  =  256,  then  125,  100,  80,  and  64  are  the 
answers. 

COMPLETE    EQUATIONS    OF    THE    SECOND    DEGREE. 
lYlt  To  solve  the  equation 

x2  -f  22)x  =  q,       (1)  Vide  §  132.      (3.) 
By  adding  j)^  to  both  sides  of  this  equation  (Ax.  IT.),  we  have 
x^  +  2j9x  +  ^2  ^  p2  _j_  ^  ^2) 

By  extracting  the  square  root  of  both  numbers  (Ax.  VI.),  we  have 

X  -{-p  =  ±:  Vp'  -f  q  (3) 

By  transposing  j^,  we  have 

X  =  —  p  ±  l/f^+q  (4) 

If  the  upper  sign  is  employed,  the  answer  is  called  the  Jirst 
root. 

If  the  loiver  sign  is  employed,  the  answer  is  called  the  second 
root. 

Xli*2»  To  solve  the  equation 

x^  —  2^)0?  ==  q.  (V) 


; 


EQUATIONS    OF    THE     SECOND    DEGBEE.       181 

By  taking  the  same  steps  as  above,  we  have 

X  =  p  ±1  Vp'  +  q.  (2) 

1'9'3.  To  solve  the  equation 

x^  -\-  2px  =  —  q.  (1) 

By  taking  the  same  steps  as  in  §  171,  we  have 

X  =  —  P  ^  Vp"*  —  q-  (2) 

X'5'4.  To  solve  the  equation  . 

x^  —  2px  =  —  q.  (1) 

By  taking  the  same  steps  again,  we  have 

x  =pdci  l/p2  _  ^^  (^2) 

ll'S.  The  four  cases  above  solved  are  comprehended  in  the 
following  general  rule  for  the  solution  of  complete  equations  of 
the  second  degree. 

RULE. 

X  is  equal  to  half  the  coefficient  of  the  second  term  taken 
with  a  contrary  sign,  db  the  square  root  of  the  square  of 
this  half  coefficient  united  to  the  second  memhcr  of  the 
equation  as  indicated  hy  its  sign. 

EXAMPLES. 

1.  Given  x^  +  4x  =  21,  to  find  x. 

Ans.  cc  =  —  2  ±  l/4  +  21,  a:  =  3  or  —  7. 

2.  Given  x"  -\-  ^x  =  20,  to  find  x. 

Ans.  cc  =  —  4  d=  t/16  +  20  =  2  or  —  10. 

3.  Given  x^  -f  lOx  =  11,  to  find  x. 

Ans.  a;  =  —  5  ±  t/25  +  11  =  1  or  —  11. 

4.  Given  x^  -|-  3ic  =  28,  to  find  x. 

Ans.  X  =  —  I  db  t/|  +  28  =  4  or  —  7. 
6.  Given  cc^  —  4x  =  21,  to  find  x. 

Ans.  x  =  2±:  1/4  +  21  =  7  or  —  8. 

16 


182   EQUATIONS  or  THE  SECOND  DLGREE. 


6.  Given  x"  —  8x  ==  20,  to  find  x. 


Ans.  a:  =  4  ±  T/IG  4-  20  =  10  or  —  2 

7.  Given  x^  —  \0x  =  11,  to  find  x. 

Ans.  X  =  5  rh  t/25  +  11  =  11  or  —  1 , 

8.  Given  x^  —  ox  =  28,  to  find  x. 


Ans.  X  =  Iziz  l^l  4-  28  =  7  or  —  4. 
9.  Given  x"^  -{-  Qx  =  —  8,  to  find  x. 


Ans.  X  =  —  S  dz  V9  _  8  =  —  2  or  —  4. 
10.  Given  x"^  -{-  8x  =  —  15,  to  find  x. 


Alls.  X  =  —  4  ±  v  16  —  15  =  —  3  or  — -  5. 
11.  Given  rc^  -f  lOx  =  —  16,  to  find  x. 


Ans.  x  =  —  6  ziz  1/25  —  16  =  —  2  or  —  8. 
12.  Given  x^  -f  5x  =•  —  6,  to  find  .t. 


Ans.  X  =z  —  ^  ziz  y  -f  —  6  =  —  2  or  —  3. 

13.  Given  x"^  —  Qx  =  —  8,  to  find  x. 

A71S.  X  =  S  dz  1-^9  —  8  =  4  or  2. 

14.  Given  x^  —  8x  =  —  7    to  find  x. 


Ans.  a:;  =  4  =fc  1/ 16  —  7  =  7  or  1. 
15.  Given  x'  —  Ux  =  —  28,  to  find  x. 


Ans.  X  =  u  ±  i/i|i    —  28  =  7  or  4. 
16.  Given  x"^  —  15x  =  —  56,  to  find  x. 

Ans.  X  =  \f  ±  -[/2|5  _  5(3  ==  8  or  7. 


17.  x^  -{-  x  =  6.     Ans.  2,  —  3. 

18.  x^  +  2x  =  8.  Ans.  2,  —  4. 

19.  a;2  +  3x  =  18.  Ans.  3,  —  6. 

20.  a;2  +  4x  =  32.  Ans.  4,  —  8. 

21.  a:'  +  5x  =  50.^«s.  5,  —  10. 

22.  a:'  -f  Gj-  =  27.  Ans.  3,  —  9. 


23.  x^  +  ISx  =  68.^ns.  4,  —  17. 

24.  x'  +  15x  =  154. 

Ans.  7,  —  22. 

25.  x^  -f  20.r  =  125. 

Ans.  5,  —  25. 

26.  .T^-f  21.r=  196.ylrjs.7,— 28. 


E  Q  U  A  T  i  0  :.'  S     OF     TILL     S  £  C  0  -N  I)     D  L  G  li  E  E.        183 


27.  x'-  —  X  ^  132. 

Ans.  —  11,  +  12. 

28.  x^  —  4:x  =  d2.Ans.  +  8,  —  4. 

29.  x""  —  Gx  =  27. 

Ans.  —  3,  +  9. 

30.  x""  —  28x  =  29. 

Ans.  —  1,4-  29. 

31.  X'  —  ox  =  —  6. 

^ns.  +  3,  4-  2. 

32.  x2  —  7x  =  —  12.  ^«s.  3, 4. 

33.  cc^  —  llx  =  —  30.  Ans.  5, 6. 

34.  a;2  —  16x  =  —  63.  Ans.  7, 9. 

35.  a;2  —  20x  =  —  96. 

.4ns.  8, 12. 

36.  x""  —  36x  =  —  320. 

Ans.  20,  16. 


37.  x^  —  38x  =  —  240. 

Ans.  8,  30. 

38.  x-"  -f  5x  =  —  6. 

^l;is.  _  2,  —  3. 

39.  x'  +  14.r  =  —  45. 

Ans.  —5,-9. 

40.  x'  +  Sx  =  —  15. 

J.92S.  —  5,  —  3. 

41.  a;2  4-  lO.r  =  —  21. 

Ans.  —  75  —  3. 

42.  x'  4-  14.r  =  —  48. 

Ans.  —  8;  —  6. 

43.  .^2  4-  20.1-  =  —  36. 

Ans.  —  18,  —  2. 

44.  x'  4-  16^  =  —  63. 

Ans.  —  9,-7. 


lt€.  All  the  equations  in  §§  171  and  174  inclusive  are  called 
reduced  equations.  Before  applying  the  rule  in  §  175,  the  given 
equation  must  be  brought  to  the  form  of  one  of  the  reduced 
equations^  by  any  process  thought  to  be  most  convenient;  that  is, 

1.  All  the  terms  involving -s?  must  he  united  in  one  terin,  ichicJi 
must  stand  first. 

2.  All  the  terms  involving  x  must  he  united  in  one  term,  icJiich 
must  stand  second. 

3.  All  the  remaining  terms  must  he  placed  to  the  right  of  the 
sign  of  equality ,  united  in  as  few  terms  as  possible. 

4.  Divide  the  whole  equation  hy  the  coefficient  of  x'. 

5.  If  then  s?  has  the  sign  — ,  change  all  the  signs  of  the 
equation. 


184       EQUATIONS    OF    THE     SECOND    DEGREE. 


EXAMPLES. 

1.  Given  -  ?^  +  22a:  -  15  =  —  -  2Sx  +  30  (1),  to  find  x. 

3  3  '         w^ 

Ojperation. 
—  5x*  4-  ^Ox  =        45  (2)  =  (1)  transposed  and  united. 

—  x^  -}-  lQx=        9  (3)  =  (2)  -r-  5. 

X   —  10a:  =  —  9  (4)  =  (3)  with  signs  changed, 

a:   =  5  ±  l/25  —  9  =  9  or  1. 

2.  Given  2x='  +  8x  +  7  =  ^  -  |-  +  197      (1),  to  find  x. 

16x2  +  64a:  +  56  =  lOo:  ^  x^  +  1576  (2)  =  (1) 

cleared  of  fractions. 

170:^  +  54x  =  1520.         (3) 

,  .  54x   1520 

x'+  —  = (4) 

17    17         ^  ^ 

27   .  ^/729   1520   _ 
X— db\ (5) 

17    ^289    17 

x=.-^    ±  —  =8  or- 11  3  (6) 
17    17 


3. 

Given  Sx^  -f  3a:  = 

530, 

to 

find  a;. 

(1) 

x^  + 

3a: 

5 

3 
10 

=b 

106 

(2) 

9 
^100 

+  106  (3) 

X  =s 

10 

or 

-  10|. 

(4) 

4. 

Sx^  —  2x  =  8. 

Ans.  2,  —  If 

6. 

2a;2  —  7a:  =  72. 

Ans.  S,  —  41. 

6. 

2a:  —  5x2  =  —  3^ 

Ans.  1,  —  -|. 

7. 

17a:  —  4a:2  =  18. 

Ans.  2,  21. 

8. 

3a;  -  21a:2  =  -  78 

Ans.  2,  —  If 

EQUATIONS     OF     THE     SECOND     DEGREE.        185 


9. 
10. 

11. 
12. 
13. 

14. 

15. 
16. 

17. 

18. 
19. 

20. 

21. 

2L. 

23. 
24. 

25. 
26. 

27. 

28. 


O.^      "Y"    ^    — —    OX. 

x^  +  12a:  —  16  =  92. 


ix 


5x2  4-  —  =  7^2  _  51, 
2 


a: 


5x 


4        24  ^^ 


5x2 


10 


X 


=  78. 

2 

14  —  2x 


6x2  _  ^  ^  92. 


22 
"9* 


4x  -  ^tl::^  =  14. 

X   +    1 
x2  -  ^  =    -    ^ 

6 
2x2  __  14^^  ^  16, 

X  H-  4       7  —  X 


=  -1  + 


4x  +  7 


X 


3x  - 


3x 


o. 


X  —  3  2 

2x2  _  30a;   +   3   =    —  X2   4-   3y3gX 

2  O  X  3 

X''   —   ox   =  --  —   -z. 

4         4 

x2  +  18x  =  —  80. 

X       -|-    Ox    -j—     J-    ^:= 

Sx 
8x2  +  -  =  -  17.^  _  34 

5 
22  —  X        15  —  X 


3 


20 
X  +  3 


+ 


X  —  6 
7x 


X  X  +  3 

24 


=  0 


f.3 

4" 


X    + 


X   ~   1 


=  3x  -  4. 


Ans.  J  or  |, 
Ans.  6  or  ~  18. 

J.ns.  6,  4|. 


J.?is.  ^,  ^, 


^?is.  4,  —  3y9^. 

J.ns.  3,  Ij. 
^?zs.  4,  —  3|. 


Ans.  4,  —  1|. 


^«s.  8,  —  1. 
Ans.  21,  5. 

^?is.  4,  —  1. 

^??s.  11,  -^Q. 
J.71S.  3,  J. 

Ans.  —  10,  —  8. 
Ans.  —  10,  — j^Q. 

Ans.  -2,-1. 

Ans.  36,  12. 

Ans.  4,  1. 

^Tis.  5,  —  2. 


16* 


186        EQUATIONS     OF     THE     SECOND     DEGREE. 

29.  -^—  =   ''  "^  ^  ■  Ans.  12,  -  2. 

X  -\-  S       2x  +  1     ■ 

30.  — ^ h  ^^±-^  =  21.  ^ns.  2,  -  3. 

X   -\-    1  X 

31.  — ' — ' —  =  —  1.  Ans.  21,  0. 

3  9  x  —  n 

32.  ?^^^  _  ^-±^  =  2.  ^«..  7,  |. 

8  —  cc  a^  —  2 

83.  — ^^ —  =  -.  ^??s.  n,  -  13. 

X  —  1       X  -{-  S       35 

34.  Given  cc'^  —  2:c  =  7,  to  find  x. 

Ans.  .T  =  1  ±  2  v'2  =  3.82842  or  —  1.82841. 

35.  Given  x"^  —  dx  =  12,  to  find  x.   Ans.  x  =  6.772  or  —  1.772. 

36.  Given  x''  —  Sx  =  20,  to  find  x.    Ans.  x  =  6.2169  or  —  3.2169. 

37.  Given  x^  +  4x  =  10,  to  find  x.    Ans.  x  =  1.7416  or  —  5.7416. 

38.  Given  llx"^  —  x  =  21,  to  find  x 

A71S.  x=  1.1412  or —  1.0824, 

39.  Given  13x='  +  2x  =  100,  to  find  x. 

Ans.  X  =  2.6975  or  —  2.8514. 

40.  Given  x""  —  Sx  =  14,  to  find  x.    Ans.  x  =  9.4772  or  —  1.4772. 

41.  Given  x^  -j-  Sx  =  8,  to  find  x.  J?is.  x  =  0.8989  or  —  8.8989 

42.  Given  x"  —  2x  =  ~  10,  to  find  x. 

A71S.  X  =  i±:  Vl  —  10  =  1  ±  3  V^^l. 

43.  Given  x^  —  20.^  =  —  104,  to  find  x. 

Ans.  X  =  10  db  2  V^^^. 


44.  Given  x"^  —  lOx  =  —  26,  to  find  x.     Ans.  x  =  5  ±  V  —  1, 

45.  Given  x""  —  12^  =  —  72,  to  find  x. 


Ans.  a;  =  6  (1  =b  V  —  1). 
46.  Given  x^  —  18x  =  —  162,  to  find  x. 

Ans.  a;  =  9  (1  db  V—  1). 

8 


47.  Given  a:  +  -  =  4,  to  find  x.         Ans.  a:  =  2  (1  d=  i    —  1). 


T  K  I  N  0  M  I  A  L     EQUATIONS.  187 


TRINOMIAL    EQUATIONS. 
fW*  A  trinomial  equation  is  of  the  form 

where  *i  is  any  number  whatever.  To  solve  such  an  equation, 
apply  the  Eule  under  §  175  for  the  value  of  x"',  after  which  the 
value  of  X  is  determined  as  in  incomplete  equations  of  the  second 


degree.     Thus,  a;«  =  p  rt  \/p^  +  q,  and  x  =V  p  ±  Vp^  +  q- 

EXAMPLES. 

1.  Given  x*^  +  4x2  ^  32^  to  find  x. 

First,      x^  ==  -  2  ±  i/4  +  32  =  — 2=h6  =  4or-8. 

Second,  x   =  d=  2  or  ±  2  V^^. 

2.  Given  x*  —  lix"  =  —  1225,  to  find  x. 

Ans.  X  =  ±  7  or  =t  5. 

3.  Given  x*  —  4x2  _  9^  ^o  find  x.     ^«s.  x  =  ±  3  or  db  l/—  1. 

4.  Given  x^  —  2x^  =  —  1,  to  find  x.  Ans.  x  =  1. 

5.  Given  x^  —  6x*  =  160,  to  find  x. 

Ans.  X  =  rfc  2  or  ±  2  l/—  1. 

I'YS.   Sometimes    an    equation    may    be    solved    by    considering 
several  of  its  terms  united,  as  the  unknown  quantity. 

EXAMPLES. 

1.  Given  (1  +  xy  -1-  (1  -f  x)  =  12,  to  find  x. 

First,  1  4-  X  =  —  -J  d=  \/\  +  12  =  3  or  —  4. 
Then  X  =  2  or  —  5. 

2.  Given  (3  -f  x^)*  —  (3  +  x'y  =  240,  to  find  x. 
First,  (3  +  x'y  =  i  ±  Vl  +  240  =  16  or  —  15. 
Then    3  -f  x«     =  ±  4  or  ±  i/^U. 

Whence        x=     =  1  or  ~  7,  or  —  3  ±  V—  15. 


X 


=  ±  1,  or  ±  1/'-  7,  or  i  l/—  3  d=  l/—  15. 


188  TRINOMIAL     EQUATIONS. 

3.  Given  (1  -f  rr  +  x^  —  2  (1  +  x  +  a:')  =  143,  to  fiad  x. 

First,    l  +  a;  +  a;2  =  ldb  Vl  +  148  =  13  or  —  11. 
Then  x"  -{■  x   =  12  or  —  12. 

,♦.     cc  ==  3,  or  —  4,  or  a:  =  -l  (—  1  =h  j/—  47). 

4.  Given  (x"  —  4x)2  +  3  (x^  —  4x)  =  0,  to  find  x. 

Ans.  X  =  4,  3,  or  1. 

5.  Given  (1  +  2x  +  a;^)^  _  ^  (1  4.  2x  +  x^)  =  254,  to  find  x. 

Ans.  X  =  3,  or  —  5,  or  —  1  dz  1/ —  15|. 

6.  Given   (x*  +  Sx^  +  16)*  —  2  (x*  +  Sx^  +  16)^  ==  389375, 
to  find  X. 

Ans.  X  =  ±  1,  or  =h  3  l/—  1,  or  db  1/  —  4  db  t/—  5, 

or  dbl/3T±~P^l^ 

7.  Given  cc*  4-  ^'  +  ^"  +  ^  +  1  =  0;  to  find  x. 

Solution. 
a-.^-f-x-fl  +7~H — ^  =  0 given   equation   divided 

X  X 

by  x"". 
a^-| --\-x  -{■  --  -f  1  =  0 last  equation  rearranged. 

a:s  -f  2  +  — +  x-{ 1-1  =  2 2,  added  to  each  side 

x^  X 

of  the  last  equation. 
|a;_|.  —  |_|.  lx-\ )=  1 last  equation  factored. 

Whence   x  -\ =  —  »  db  1  t/5 

X  z         ^ 

a;  =  -  1(1  rip  t/^  q=  V  -  10  zp  2i/5). 

8.  Given  x*  —  x'  -f-  x''  —  x  +  1  =  0,  to  find  x. 

Am.  X  =±:  I  (1  ±  1/5  d:  1/—  lU  d=  2  y^5  ). 


LITERAL     EQUATIONS.  189 

1*79.  LITERAL    EQUATIONS. 

1.  Given  x   —  2ax  =  2ab  -f-  ^^  to  find  x. 

Ans.  x=  azt:  V  a'  +  'lab  +  V  =  a  ±  (a  +  5)  =  2a  +  &  or  —  6. 

2.  Given  x"^  —  2ax  =  —  a"^  +  h"^,  to  find  x. 

Ans.  X  =  a  zt:  y  a}  —  d^  •{•  1?  =  a  dcz  h. 

3.  Given  x^  —  (<^  +  ^)-c  =  —  ah,  to  find  x. 

2^4  2 

a  —  h  , 
=  a  or  6. 

2 

4.  Given  ct^  —  (a  —  V)  x  =  «,  to  find  x.    Ans.  x  =  or,  or  —  1. 

5.  Given =  r,  to  find  x.    Ans.  x  =  h  dc  ^Z ab  -\-  I?. 

X  -\-  a        x  —  6 

6.  Given —  h  = ,  to  find  x. 

X  —  a  a  -\-  X 

aVb  +  2 


Ans.  X  =  da 


(J        an  -V  ff.  !  '/* 

7.  Given -j-  5  =  "  ,  to  find  x. 

a  -\-  X  a  —  X 


1/6  —  2 


Ans.  a:  =  ^  (—  2  ±  1/4  +  Z/^). 


^I'X 


8.  Given  a?  +  6^  _  9ix  +  a;^  =  ^—  (1),  to  find  x. 

ahi^  +  6V  _  26?i2x  4-  ?i'x2  =  m^^-,     (2)  =  (1)  cleared  of  fractions. 
nKz""  —  m^x^  —  2bn^x  =  —  ahi''  —  bhi"  (3)  =  (2)  transposed. 

^,__    m^  _  -  ^'^^  -  f  ^'  (4)  =  (3)  -  (n^  -  mO. 
Then  x  =  -^ i'^'\-n ;^ + 


X 


n^  —  m^        ^  (?i^  —  m^)^  n^  —  ni^ 


bn^  rb  n  \/ ahn"^  4-  bhu"^  —  a^ 

xz= 

n^  —  iw 

n 


And  finally,  x  =  — ,  (bn  db  \/ a^m''-  -f  ¥m^  —  a^n?'). 


{Vid^^  S8,  ex.  16.) 


190  RADICALEQUATIONS. 

EQUATIONS    CONTAINING   RADICAL    QUANTITIES. 
180.  Equations  containing  radical  quantities  are  usually  solved 
by  a  judicious  application  of  Axiom  VII.     No  rule  can  be  inva- 
riably followed  in  such   equations.      Some   general   directions  will 
be  better  understood  after  tlie  solution  of  a  few 

EXAMPLES. 

1.  Given  7  l/x  +  5  =  10  +  4  i/^  (1),  to  find  x. 

3  t/x  =  5.  (2)  ==  (1)  transposed  and  united. 

Vx  =  |.  (3)  =  (2)  -  3. 

X  =  V.  (4)  =  (3)3.         (Hc?e  Ax.  VII.) 

2.  Given  Vx  —  8  =  Vx  —  V'2>  (1),  to  find  x. 

a:  _  8  =  X  -  2l/2^+  2.     (2)  =  (1)^  Ax.  VII 
l/2x"=  5.  (3)  =  (2)  reduced. 

2x  =  25.  (4)  =  (3)^ 

Whence        x  =  12i- 


3.  Given  -  +  ^   ^l^^'  ==  -  (1),  to  find  x. 
^  X  4 

|/16  -  x^  _  ^  __  4  ^2)  =  (-1)  transposed. 


X 


4        X 


Whence  x  =  ±  4.  (4)  =  (3)  reduced. 

4.  Given  }^E±1  +     ,_!_  =  -A^  (1).  ^o  find  x. 

l/x    _      l/x  +  9       |/x  +  9 

X  +  9  +  6  l/x  =  4x  (2)  =  (1)  cleared  of  fractious. 

6i/x  =  3x  —  9.  (3)  =  (2)  transposed. 

36x  =  9.x2  —  54x  +  81.  (4)  =  (3)  squared. 

Whence   x  =  9  or  1.  (5)  =  (4)  reduced. 

_     „.  x  —  9  X  — 4         4Cx  —  16)  .-,.    .     .    , 

5.  Given      ,-    ,    »  H 7= ^  =        -    ,    /   (1),  to  find  x. 

■y/x  +  3        ^/x  —  2  |/x  H-  4    ^  '^' 

(  Vide  §  163,  ex.  32.) 


RADICAL     E  Q  U  A  T  I  0  ^*  S.  191 

|/^  __  3  4-  y'x  +  2  =  iV'x  ~  16.   (2)  =  (1)  with    each    term 

reduced. 
2  y'x  =  15.  (3)  =  (2)  with  terms  united. 

Whence  x  =  56|  (4)  =  (3)^  and  reduced. 

^ 2a2 

6.  Given  x  +  V  a'  +  x'  =     .--- -„  (1),  to  find  a:. 

V  «    +  it- 

£C  i/a^  +  a;2  +  «2  ^  .^2  _  2a2.  (2)  =  (1)  cleared  of  fractions. 

X  Va"  +  x^  =  «'  —  x^  (3)  =  (2)  transposed. 

aKv""  +  x^  =  «^  —  2a2x2  +  x'  (4)  =  (3)2. 

Sa^x^  =  «*  (5)  =  (4)  reduced. 

Whence     x  =  ±  -  -1/3. .  (G) 

o   *^ 


1.  Girea  V^+f^  +  -r^^  =  -^^  (1),  to  find  x. 
-j/x  V X  -\-  a        V X  -\-  a 

"  +  %2V^  =  J'  (2)  =  (1)  X  ^^  +  ~" 


X 


X  i/x 


1  _!.  11  _{_  2^/"  =  Z/2  (3^  =  (2)  modified. 

X  \  X 

^  4.  2-^/"  +  1  =  6'      (4)  =  (3)  modified. 

X  \  X 

142  +  1  =  ±  ^  (5)  =  1^(4). 

Wx  =  J^.  (6)  =  (5)  reduced. 

•^  6  zp  i 

W  hence    x 


^     ^.  l/x  +  l/x  —  a  ^r-^a      /IN    ,     £    J 

8.  Given  ^     /-=^  =  "  ^1)?  to  find  x. 

y  X  —  y  X  —  a        X  —  a 

(y^  -i-l/x-  ay  ^     ^^'^^    .(2)  =  (1)  modified  (TWe 
a  X  —  a 

§  164,  ex.  9). 

y^  _^  y^x  —  a     =  — =i^.  (3)  =  (2)  modified,  and 

y^x  —  a 

root  taken. 


192  RADICAL    EQUATIONS. 

l/x^  —  ax  -{-  X  —  a  =  dz  na         (4)  =  (3)    cleared     of 

fractions. 

l/x^  —  ax  =  a  (1  ziz  n)  —  X  (5)  =  (4)  transposed. 

x^  —  ax  =  a\l  ±  7iy  —  2ax  (1  ±  n)  +  x^  (6)  =  (5)^ 

Whence  x  =  —z. — -, — r^-^- 

1  ±  Zn 

From  the  exam^Dles  now  given,  it  will  have  been  seen  that  the 
object  has  been,  in  every  instance,  to  relieve  x  from  its  radical 
sign,  after  which  its  value  is  obtained  in  the  usual  way. 

To  effect  the  object,  the  terms  of  the  equation  must  be  so 
arranged  that,  on  squaring,  as  many  of  the  radicals  as  possible 
will  disappear. 

If,  on  squaring,  radical  terms  still  remain,  re-arrange,  and  square 
the  equation  a  second  time. 

Examples  7  and  8,  above,  exhibit  anomalous  methods  of  solu- 
tion. They  should  be  carefully  studied, — that  is,  studied  until  the 
reason  for  each  change  is  clearly  perceived. 

The  pupil  will  find  in  the  following  examples  ample  oppor- 
tunity to  improve  his  powers  of  analysis ;  and  we  take  this  occa- 
sion to  remind  both  teacher  and  piqnl,  that  a  day  occupied  in 
the  investigation  of  a  single  equation  is  discreditable  to  no  one 
desirous  of  obtaining  a  familiar  acquaintance  with  the  various 
operations  of  algebra.  Indeed^  such  examinations  are  absolutely 
necessary/. 

Complicated  equations  can  generally  be  solved  in  a  variety  of 
ways,  but  the  best  method  can  be  learned  only  from  practice. 

As  a  further  illustration,  we  will  resume  example  4  above,  and 
then  leave  the  pupil  to  exercise  his  own  ingenuity. 

9.  Given  "^^^^  +       ±_   =  _±g^  n^  to  find  x. 

y^X  Vx   -\-d  Vx   +    9   ^ 


RADICAL    EQUATIONS.  193 

?  -I-  _£_  4_  1  =  4.       (3)  =  (2)  modified. 

Whence  a;  =  9  or  1. 

From  (4)  we  liave  l/x  =  3,  or  l/x  =   —  1 ;  and  it  is  with 
this  limitation  that  the  value  1  satisfies  the  original  equation. 

10.  Given  17  +  2  Vx^  +  9  =  27,  to  find  x.    Ans.  x  =  ±  4. 

11.  Given  5  —  l/25  —  x'^  =  Sx,  to  find  x. 

12.  Given  \/x  —  82  =  16  —  Vx,  to  find  x. 

13.  Given  Vx  +  40  =  10  —  Vx,  to  find  x. 

14.  i/x  —  16  =  l/x  —  2. 

15.  i/^+~8  —  t/^^^:^  =  2  i/2. 

/  / 9 

16.  yx-i-yx  —  d=  ^- 

y  a:  —  y 

*17.   l/l  +  X  Vx""  —  1  =  1  —  a:. 

l/x  +  1/^10  —  x 


18.  i/x  —  i/lO  —  X  = 


2 


^«s 

t.    X  = 

:    3. 

J.71S. 

X  = 

81. 

J.ns.  X  = 

=  9. 

Ans. 

25. 

Ans. 

10. 

Ans. 

12. 

Ans 

5 

•     4* 

Ans 

t.  9. 

9x  —  1        ,    .    l/9x  —  1  .       o 

19.  — ,= =  4  + Ans.  9. 

l/9x  +  1  2 

20.  Sv'g+lO  _  l/2x  +  16  ^^^^^    4, 
3  i/2x  —  10        l/'2x  —  4 

21    V:^+^  =  VUl^.  Ans.  4. 

1/:^  +  4         i/x  +  6 

22.  -  +  ^""^  ~~  ^'  =  ^.  ^71.9.  ±  V2ab  -  h\ 

X  X  0 

^^„    2t/x  +  a  2a  —  i/x  ^  2  ^    ^^«' 

*23.  —i^ ■ = Ans.  a*  or  —77-- 

yx  +  2a  -j/x  9 

a  —  Va'  —  x2        ,  .        _;_  2a  l/t" 

24. —  =z  b.  Ans.  ± 

a  -f-  1    o2  __  ^2  1   +  6 

n 


1 D4  P  R  0  L  L  E  M  ?. 


^  -    Va   +   X        V  a  —  X  \/x  ,  .    ^     / 

yx  .  yx  yi) 

_^    a  +  .T  -f  V'2>ax  +  a;2  ±  a  (1  ±  i/26  —  Z;^) 

ZD. =  6.  ^?2S. ^^ i^- 


5  -f  i/25   —  a;2        ^ 

l/8  +  a^        l/8  -  ;:c        Vx 

*28. j — —  =  — -.  Ans.   =b  8. 

yx  y  X  J 

*29.  1^^  +  V5  =  l^-ii^.  ^«..  2  or  _  10, 

l/20  4-  .T  l/20  —  :c 

,„    l/4x  4-  20       4  -  t/x  64 

dU.  -i — , -^  = y= — .  A?is.  4  or ^• 

4  +  \/x  -/x  3 

l/x  4-  1       t/x  —  1  ^        .  « 

ol.  4 .  =  a.  ^?is.   ±  • 

y a;  —  1       yx  4-  1  va^  —  4 

32.  l/a^  -\-  ax  ==  a  —  ya^  —  ax.  Ans.   db  9  1/3. 


33.  l/a  4-  t/x  =  l/a^;.  ^?2S.  J 


a 


(Va-iy 

PROBLEMS 
INVOLVING   EQUATIONS   OF    THE    SECOND   DEGREE. 
181.  1.  Three   times   the   square    of   a   number   added    to   four 
times  the  number  is  equal  to  64.     What  is  the  number? 

8a;2  4-  4x  =  64.  Ans.  4  or  —  5J. 

2.  A  man  bought  a  number  of  sheep  for  $200,  and,  reserving 
20,  he  sold  the  remainder  for  $150,  gaining  $1  on  the  price  of 
each  sheep.     What  number  was  purchased? 
Let  X  =  the  number. 

Then   —  =  price  per  head  of  those  bought. 

and p-pr  =  price  per  head  of  those  sold. 

X  —  20 


P  R  0  B  L  E  M  S.  195 

Now,  by  the  question  tlie  latter  price  is  $1  more  than  the 
former.     Hence, 

—  -f  1  =  — -.  An^.  50  sheep. 

X  X  —  20 

3.  Divide  50  into  two  parts  so  that  their  product  may  be  621 

An^.  23  and  27. 

4.  The  difference  of  two  numbers  is  6,  and  their  product  is 
216.     What  are  the  numbers  ?  Ans.  12,  18. 

5.  A  man  sold  a  watch  for  $75,  and  gained  as  much  per  cent, 
as  the  watch  cost  him.     What  did  he  pay  for  it?  Aiu.  $50. 

6.  A  man  sold  a  watch  for  $24,  and  lost  as  much  per  cent,  as 
the  watch  cost  him.     What  did  he  pay  for  it  ?         Ans.  $40  or  $60. 

7.  If  7  be  added  to  a  certain  number  and  3  be  subtracted,  the 
product  of  the  sum  and  difference  will  be  119.  What  is  the 
number?  Ans,  10  or  —  14. 

8.  A  merchant  bought  a  quantity  of  flour  for  $72.  Had  he 
bought  6  barrels  more  for  the  same  sum,  the  price  per  barrel 
would  have  been  $1  less.  How  many  barrels  did  he  buy,  and 
at  what  price  per  barrel  ?  Ans.  18  barrels,  at  $4  per  barrel. 

9.  If  a  certain  number  is  subtracted  from  12  and  the  remainder 
is  multiplied  by  the  number,  the  product  will  be  35.  What  is 
the  number  ?  Ans.  5  or  7. 

10.  If  a  certain  number  be  divided  by  10,  and  this  quotient 
be  added  to  the  quotient  of  10  divided  by  the  number,  the  sum 
will  be  34.     What  is  the  number?  Am.  30  or  34. 

11.  A  man  travelled  105  miles,  and  then  found  that  if  he  had 
gone  2  miles  less  per  hour  he  would  have  been  6  hours  longer 
on  his  journey.     At  what  rate  did   he  travel  per  hour? 

Ans,.  7  miles. 

12.  Divide  40  into  two  parts  so  that  the  sum  of  their  squares 
may  be   1000.  Ans.  30  and  10. 

13.  Two    fields    differing    in    quantity   by    10    acres   were    each 


196  PROBLEMS. 

sold  for  $2800,   one  bringing   $5  per    acre    more    than  the  other. 
What  was  the  number  of  acres  in  each  ?      Ans.   70  and  80  acres. 

14.  The  product  of  two  numbers  is  120.  If  2  be  added  to 
the  less  and  3  be  subtracted  from  the  greater,  the  product  of 
the  sum  and  difference  will  still  be  120.  What  are  tlie 
numbers  ?  Ans.  8  and  15. 

15.  Two  men  are  travelling  towards  each  other.  On  meeting, 
B  has  travelled  20  miles  farther  than  A.  A,  bj  preserving  his 
rate  of  travel,  will  go  the  distance  B  has  already  travelled  in  20 
hours;  but  B  will  be  only  15  in  passing  over  A's  distance  (at 
his  former  rate).     What  is  the  rate  per  hour  of  each  ? 

Ans.  A,  7.464,  and  B,  8.61  jf. 

16.  Two  merchants  sold  the  same  kind  of  stuff,  and  together 
received  $35.  The  second  sold  3  yards  more  than  the  first. 
Had  the  prices  per  yard  been  interchanged,  the  first  would  have 
received  $24  and  the  second  $12J,  gaining  thereby  $1^.  How 
many  yards  were  sold  by  each,   and  at  what  price  per  yard? 

Ans.  15  at  $1-J  and  18  at  $|,  or  5  at  $3  and  8  at  $21. 

17.  Divide  the  number  10  into  two  parts  so  that  10  times 
the  second   part  may  be  the  square  of  the  first  part. 

Ans.  5  (—  1   +  1/5)  and  5  (3  —  l/5.) 

18.  Divide  the  number  a  into  two  parts  so  that  the  square  of 
the  second  part  may  be  the  first  multiplied   by  a. 

Ans.   -  (3  ±  i/5)  and  ^  (-  1  =f=  i/5)- 

19.  A  and  B  travel  at  the  same  rate  towards  Washinoton.  At 
the  50th  mile-stone  from  Washington,  A  overtakes  a  flock  of 
geese  travelling  1^-  miles  an  hour,  and  two  hours  afterwards 
meets  a  coach  travelling  2\  miles  per  hour;  B  overtahes  the  geese 
at  the  45th  mile-stone,  and  meets  the  coach  40  minutes  before 
reaching  the  31st  mile-stone.  What  is  the  distance  between  A 
and  B?  Ans.  25  miles. 


EQUATIONS    VriTIl    TWO    U  N  K  N  0  W  X    Q  U  A  X  T  I  T  IE  S.  1P7 

EQUATIONS    WITH    TV.'O    UNKNOWN    QUANTITIES. 

182.  1.   Given  .r  +  7/  =  8     (1),  ]  ^     .    ,  , 

y  to  nnd  x  and  y. 
and  cci/  =  Id  (!^),  j 

x^  +  2x^  +  ^^  =  64,  (3)  =  (1)=. 

4r^  =  60,  (4)  =  (2)  X  4. 

x^  -  2x1/  +  if  =  ^,     (5)  =  (3)^  (4). 

x   -      y  =  2,     (6)  ==  Vip). 

X  =  5,     (7)  =  ((!)  +  (6)  )  -  2. 

y  =3,  ((8)  =  (l)-(6))-2. 

The  above  operation  v/ill  be  readil}^  understood,  and  tlie  object 
of  eacli  step.     In  the  same  way  solve  and  verify  the  following: — 

^   (x-+^=10.)  ^    (a;+^=12.)  ^     (^+i/=20.)  ^     (x+y=50.  ) 

•^'  (     ^-^=16.)  ""•  (     .T^=32.)  •   (     .r^=:64.)  •   (     ^-^'=400.) 

(6.)                       (7.)  (8.)  (9.) 

(a:+^=3-i.)       (.r+^=-l.  )  {x+I/=      14.)  (a;+y=     2.  ) 

(     ^i.'=U-)      (     .'^•J/=-56.)  (     .r^=+45.)  (     .Ty=~63.) 

1S3.  (1.)  Given  x  -\-  y  =  a  (1),  ]  ^     _    , 

y  to  find  X  —  y. 
and  .ry  =  h   (2),  j  -^ 

X'  +  2^-y  +  ^=  =  «^  (3)  =  (1)^ 

\xy  =  4Z>,(4)  =  (2)  X  4. 

x^  -  2xy  +  ^^  =  a^  -  45,  (5)  =  (3)  -  (4). 

X  —  y  =  Va^  —  46,  (6)  =  l/'(5). 

Hence,  when  the  sum  and  product  of  two  numbers  are  given, 
take  the  square  root  of  four  times  the  product  subtracted  from 
the  square  of  the  sum,  and  this  root  will  be  the  difference  of  the 
numbers. 

(2.)  Given  x  -\-  y  =  10,  and  xy  =  24,  to  find  x  and  y. 
By  the  rule  x  —  y  =  j/lO^  —  4  x  24  =  2. 
Hence,  x  =  6  and  y  =  4. 


198  EQUATIONS    WITH 

(3.)  (4.)  (5.)  (6.) 

(x+y  =  13.)      Cx--f^  =  n.)       (x+y  =  20.)       (a) +  3.  =  5^.) 
(      a-y=24.)       (      2;y  =  28.)       (      xi/  =  dQ.)       (      ^3/ =  2^0 

184.  1.  Given  x  -  7/  =  S    (1),  -)   ,     .    ,  , 

V  to  una  X  and  y. 
and  xi/  =  48  (2),  J  "^ 

x^  -  2:.-y  +f^  64,     (3)  -  (1)1 

4xy  =  192,  (4)  =  (2)  X  4. 

:,2  j_  2;,.^  +  y  =  256,  (5)  =  (4)  +  (3). 

X  +  7J  =16,    (6)  =  i/(5). 
Hence     cc  =  12  and  ^   =  4. 

In  the  same  manner  solve  the  following  equations. 

(2.)  (3.)  (4.)  (5.) 

(x  —  ?/  =  4.  )  (.X  —  ?/  =  3.  )    (x  —  y  =  5.     )   (x  —  ?/  =  3^) 

(       x^  =  21.)  (       x^  =  70.)    (       xj/  =  300.)   (       x^  =  2.  ) 

(6.)  (7.)  (8.)  (9.) 

(.T  -  y  =  -  2.)   (X  -  y  =  7~.)   (x  -  y  =  1.  )   (x  -y  =  11.) 
C       a^  =      24.)   (        x^  =  2f .)   (        .;y  =  3|.)   (       :ry  =  26.) 


10.  Given  x  —  y  =.  a  (1), 
and  xy  ■=  h  (2), 


I  to  find  .T  +  y- 


X 


2xy  -\-f  =  a\  (3)  =  (1)^. 


^xy   =  \h,  (4)  =  (2)  X  4. 
x'  +  2x2/  4-  2/'  =  «^  +  45,  (5)  =  (4)  ±  (3). 


X  +  ;?/  =  /ct^  +  46,  (6)  =  T/C5). 

Hence,  when  the  difference  and  product  of  two  numbers  are 
given,  take  the  square  root  of  four  times  the  product  added  to 
the  square  of  the  difference,  and  this  root  will  be  the  sum  of 
the  numbers. 

11.  Given  x  —  _?/  =  3,  and  xy  =  28,  to  find  x  and  y. 


By  the  ride,         x  +  y  =  V  9  +  28  X  4  =  11. 
Hence  x  =  7  and  y  =  4. 


TWO     U  N  K  N  0  W  N     QUANTITIES.  199 

(12.)  (13.)  (14.)  (15.) 

(a:  -  y  =  10.  )    (.x  -  y  =  5.  )    (x  -  y  =  21.)    (x  -  ^  =  5^.) 

(       x^  =  119.)    (        X2/  =  24.)    (        x^  =  U.)    (       rry  =  3.  3 

185.  1.  Given  a;^  +f  =  25  (1),  ]   ^     .    .  , 

^  to  nnd  X  and  y. 
and      X   -\-  ^   =  7    (2),  ) 

^2  _|_  2^.^  +  ^2  _  49^     ^3^  _  ^2)^ 

2:ry  =24,     (4)  =  (3)-(l). 

x^  -  2xy  +  y  -=  1,       (5)  =  (1)  -  (4). 
x-y   =1,       (6)  =  1/(5). 
Hence  a:  =  4  and  ?/  =  3. 

In  the  same  way  solve  the  following  equations. 

(2.)  (3.)  (4.)  (5.) 

x"  +  if  ^  50.      x^  +  ?/2  =  5.      .t-  +  y'  =  29.      x""  -\-  if  =^  40. 
a:    +  y   =  8.         a:    +  y   =  3.       a-    +  y   =  7.         x   -\-  y   =  ^. 

6.  Given  a;    +  y   =  a  (1),  )   ,     ^    ,  , 

y  to  nnd  X  and  ?/. 
and       a;=^  +  /  =  c  (2),  j 

^2  _f_  2xij  +f  =  a\  (3)  =  (1)^ 

2xy  =  a^  -  c,  (4)  =  (3)  -  (2). 

X'  -  2xy  +  f=2c-  a\  (5)  =  (2)  -  (4). 


x—y  =  V-lc  —  a\  (6)  =  1/(5). 

Hence,  when  the  sum  of  two  numbers  is  given,  and  also  the 
sum  of  their  squares,  take  the  square  root  of  the  square  of 
the  sum  subtracted  from  twice  the  sum  of  their  squares^  and  this 
root  will  be  the  difference  of  the  numbers. 

7.  Given  x"^  -{-  y"^  =  89.  and  x  -\-  y  ^  13,  to  find  x  and  y. 
By  the  rule,  x  —  y  =  l/89  X  2  —  13*^  =  3. 
Hence  a:  =  8  and  y  =  b. 

(8.)  (9)  (10.)  (11.) 

(x3+y2=50|.)     (x^^y^=2b.)     (^2+/=     58.)     (x^-^y^=     41.) 

(x+y=.10.  )     {x+y=7.)     (x-{-y=^10.)     (.x+y=-l.  ) 


[  to  find  X  and  y. 


200  E  Q  U  A  T  I  0  N  S     AV  I  T  11 

(12.)  (13.)  (14.)  (15.) 

(x'+r=nOb.)  (...2-f/=34.)       (x^+?/2=65.)  (^•2+y=10.) 

(.r  -i-y  =47.     )  (r.  +^  =2.  )       (X  +y  =9.  )  (x  +j/  =3.  ) 

186.  1.  Given  x-"  +  f  =  52  (1), 

and       X   —  9/   =  2    (2), 

X'  +  2xy  +2/^  =  4,  (3)  =  (2)^ 

2xij  =  48,        (4)  =  (1)  -  (3). 

x-^  4-  2.r^  +  ?/^  =  100,      (5)  =  (1)  +  (4). 
+  y    =  rh  10,   (6)  =  t/(5). 
Hence        a;  =  6  or  —  4,  and  y  =  4  or  —  6. 

In  the  same  wr.y  solve  the  following  equations. 

(2.)  (3.)  (4.)  (5.) 

(a;=+/  =  25.)   (.x-^  +  7/^  =  41.)   (x2+y^  =  65.)   (x2  +  y^=61.) 
(03   —y  =  1.   )    (:c   —  3^  =  1.  )    (a?  —  y  =  3.  )    (x  —  y  c=  1.  ) 

C.   Given  :r    —  i/   =  d  (1), 
and       x^  -}-  ?/2  =  c  (2), 

.t''  —  2X2/  -\-  1/"^  =  iV 

2xy  ^  c  —  iV" 

X-  4-  2x\j  -^  if  ^2c  ^  d"" 

X  +  y   =  ±  -l/2c  —  c?2. 

Hence,  when  the  difi'erence  of  two  numbers  is  given,  and  also 
the  sum  of  their  squares,  take  the  square  root  of  the  square  of 
the  difference  subtracted  from  twice  the  sum  of  their  squares, 
and  this  root  will  be  the  sum  of  the  numbers. 

(7.)  (8.)  (9.)  (10.) 

(.,;= -f  3/2  ^  74.)    (x2  + 7/2  =  45.)     (x2  4-?/2  =  65.)     {x^  -\-y' =^^h:) 
{^x  -7/  =2.  )    {x  -y  =3.  )     (X  -y  =3.  )    {x  -y  =11.) 

187.  1.  Given  x'  -  y»  =  17  (1),  )   ^     .    , 

r.     ,r^     ^  to  find  X  and  ?/. 
and       X  -y  ^2    (2),  j  *^ 

^-^+^  =  8^,(3)  =  (1)  -?-(2.) 

Hence  .t  =  5|,  and  y  =  3|. 


>■  to  find  X  and  y. 


T  W  0     U  ^*  K  N  6  W  N     QUANTITIES.  201 

In  the  same  \Yay  solve  tlie  equations — 

(2.)  (3.)  C'l-)  (^0 

(,,^  -f=.  55.)    {X-  -  3/^  =  12.)  {x^  -  y  =  IB.)  {X-  -f=^  14.) 
(x  -rj  =11.)   (x  -y  =3.  )  (X  -y  =U.)  (X  -y  =5.  ) 

(6.)  (7.)  (8.)  (9.) 

(^,.2  _y.^  15.)    (x^  _  ^2  =  26.)  (x^  -  y  =  15.)  (x^  -  3/'  =  30.) 

(a;  -y  =10.)    (X  -y  =13.)  (x  -i/  =3.  )  (x  -y  =60.) 

188.    1.  Given  x^  -  r  ==  ^^  (1),,  |  ^^  ^^^  ^^  ^^^  ^^ 
and      X   +  y   =  15  (2),  j 

a-_y  =  l,    (3)  =  (1)  -^  (2). 
Hence  cc  =  8,  and  ?/  =  7. 

In  the  same  way  solve  the  equations — 

(2.)  (3.)  (!•)  C^O 

(^2_y_18.)     (x2~7/-^=     27.  )      (x^-/=53.  )     (x^-7/^=500.) 

(:,+y=9.  )     (x+y=-13^.)      (:r+j/=17f.)     (x -y  =125.) 

6.  Given  x^  -  f  =  -.  |  ,,  g.^  ,  ,,a  y. 
and      X   —  y  =  d  .) 

in  +  d^  wi  —  fZ2 

7.  Given  x^-,/=^  m,  |  ^^  ,^^  ^  ^_^^  ^^ 

and       X   -{■  y   =  a,  ) 

^2  -f  m  «'  --WI 

189.   1.  Given  x^  +  ^  =  5  (1),  |  ^^  ^^^  ^  ^^^  ^^ 
and  x?/  =  2  (2),  j 

2xy  =        4,     (3)  =  (2)  X  2. 
^2  +  2xy  +  y^  =        9,     (4)  =  (1)  +  (3). 

cc^  -  2xy  +  3/'  =        1>     (^)  =  Cl)_r  ^'^^* 
cc  +  ?/   =  ±  3,     (6)  =  1/(4). 

X  _  y   =  ±  1,     (7)  =  l/(5). 
Hence  a:  =  ±  2  and  3/  =  ±  1. 


202 


EQUATIONS    WITH 


In  the  same  way  solve  the  equations — 

(2.)  (3.)  (4.)  -         (5.) 

(0^2+^=10.)    (^x'-\-f=is.')    (x'^f=.m:)    (xH^'=   13 ) 

(      icy=3.  )      (     Ty=6.  )      (      ^^=8|.  )       (      o-y=~6.  ) 
(6.)  (7.)  (8.)  (9.) 

(     a:^=-24|.)     (      cr?/=-7|.)    (     o^y  =25.)    (      ^3/ =12.) 

10.  Given  x^  +  j/'^  =  c,  and  .t^  =  h,  to  find  a;  and  y. 


±i/c-  -f  26  ±t/c— 2&           rbl/c  +  26=b-i/c  — 2i 
^ws.  a  = ^ ,  7/  =t  — . 


190.  1.  Given  x^  —  y  =  7    (1), 


to  find  X  and  y. 


rry   =  12  (2), 

x^  —  2xy  +  y  =       49,  (3)  =  (ly. 

4:xy  =        576,  (4)  =  4  X  (2)2. 

x'^  +  2x^2  +  y*  =        625,  (5)  =  (3)  +  (4). 

x^  +  f  =  zh  25,  (6)  =  t/(5). 


Hence   a:  =  ±  4,  or  rt  3  V  —  1,  and  y 


±4i/=rT. 


In.  the  same  way  solve  and  verify  the  following  equations. 

(2.)                       (3.)                        (4.)  (5.) 

(£c2— y2^24.)  (:x«— y2  =  21.)    (^2— y»=16.)  (a;2— y2^40.) 

(        ^_y  =  5.  )  (        ay  =  10.)    (        xi/  =  15.)  (       xi/  =  21.) 

(6.)                     (7.)                       (8.)  (9.) 

(x2_y=60.)  (^;2— y2=80.)        (x"—y=l.     )  (x^— y2=3.        ) 

(      XT/  =16.)  (      :ry  =9.   )        (       x?/  =t/6.)  (      xij  =l/lO.) 


i2i57  7/^Tr. 

191.             (1.) 
x""  —  f  =  19. 

X2 

(2.) 
-  y  =  16. 

X   +  y   =19. 

Ans.  X  =  10,  y  =  9. 

3; 

-y  =2. 
Ans.  X  = 

'^;y 

r-3. 


TWO     U  N  K  N  0  Vv'  N     QUANTITIES. 


203 


(3.) 

(4.) 

ic2  —  /  =  15. 

c«2  _  ^2  ^  120. 

a:  —y   =  3. 

cc    —  y   =  10. 

Ans.  X  =  4,  1/  =  1. 

Ans.   X  =  11,  y  =  1. 

(5.) 

(6.) 

x^  ^7f  =  39. 

x^  —  ^^  =  6. 

^  +  y  =13. 

X    -\-  y   =^. 

A71S.  X  =  S,  y  =  5. 

Ans.  cr  =  31,  y  =  2|. 

(7.) 

(8.) 

a;2  —  ?/2  ^  40. 

_^2  _  y  =^  45. 

X    -\-  y    =10. 

.^   +  y   =9. 

J.WS.  a;  =  7,  3^  ^  3. 

J.7ZS.  cf  =  7,  y  =  2. 

(9.) 

(10.) 

x+y  =  8. 

cc  —  y  =  1. 

cry  =  15. 

xy  =  6. 

Ans.  X  =  5,  ?/  =  3. 

Ans.  .X  =  3,  y  =  2. 

(11.) 

(12.) 

a;2  -f  y2  =  169. 

a-2  —  y2  =  16. 

X   -\-  y   =Vl. 

a;    +  y   =  8. 

Ans.  X  =  12,  7/  =  5. 

Ans.  cc  =  5,  y  =  3. 

(13.) 

(14.) 

^2  _  y  ^  21. 

x'  +  y'  =  325. 

ir   —  y   =1. 

c^y    =  150. 

J.7iS.    CC   =    11,  y   =    10. 

Ans.  cc  =  15,  y  =  10. 

(15.) 

(16.) 

x^  —y""  ==  33. 

x"  -\-  y""  =  1300. 

cry    =  272. 

cc  —  y  =  10. 

ns.  cr  =  db  17,  or  =h  16  V —  1, 

^7is.  CC  =  30,  y  =  20. 

y  =  ±  16,  or  ±  17  V—  1. 

• 

(17.) 

(18.) 

x'-^f   =    lyi^. 

X'  -f=-  9. 

^y  =  -  i- 

X    —  y    =  —  1. 

^??s.  a;  =  1,  y  =  —  |. 

Ans.  X  =  4,  y  =  5. 

204 


EQUATIONS     AV  I  T  n 


(19.) 
X    —  y    =1. 

Am.  X  =  \  -^  l/5, 

^  =  1/5  —  i- 
(21.) 
0^  -1-  ^2  ^  44. 

.Ty  =  3  1/51. 

J.71S.  X  =  ±  3  V  3,  or  ±  j/lL 

t/  =  ±  i/lT^  or  rt  3  VZ. 

(23.) 
a;2  +  7/2  ^    123. 

X    —  y   1/3. 

^  =  4  1/3; 

192.  1.  Given  x""  -j-  y  =  9  (1), 
and       :«    +  7/   =  3  (2), 


(20.) 
o;^  -3/^  =  1. 


xy 


=  9. 


1/3. 


u.4?is.  a-  =  =fc  2,  or  d=  y  —  3 
2^  =  =fc  1/  3,  or  ±  2i/—  1. 

(22.)       . 
a;2  +  7/2  =  26. 

•-c    —  y    =  1/2. 

J.?is.  X  =  3  1/2. 

X  =  2  1/2^ 

(24.) 

0^2  _  y  =  45^ 

X  —  y  =  |/^5. 

Ans.  X  =  D  Vb. 

y 

to  find  X  and  2''' 


=  4i/5, 


FIRST    METHOD. 


X»  +  Zx-^y  +  Sx/  +  3,'3  _  27, 

3xV  4-  3x^2  ^  i§^ 

^y  (^  +  ^)  =  6. 


xy  =  2. 


X  ^  y 

x 


(3)  =  (2)^ 

W  =  (3)  -  (1). 

(5)  =  (4)  -f-  3  and  factored. 

(6)  =  (5)  -  (2). 
Vide  5  183. 


Hence  x  =  2  and  ?/  =  1. 

In  the  same  M'^ay  solve  and  verify  tlie  following  equations. 
(2.)     .  (3.)  (4.)   •  (5.) 

(x'+3/'  =  35.)    (a;'+y''  =  91.)   (x^  +  ^' =  341.)  (x»+y  =  65.) 
(x  -\-y  =5.  )    (x  +7/  =7.  )   (x  +y  =11.  )    (x  +  y  =5.  ) 

X  =  3,  y  =  2.      X  =  4,  ?/  =  3.     x  =  G,  y  =  5.     x  =  4,  ?/  =  1. 


TWO     UNKNOWN    QUANTITIES.  205 


193.  1.  Given  x'  +  f=^d  (1), }  ,     .    ,  , 

^  to  una  X  and  y. 
and       X   -\-  y   =3  (2),  ) 


SECOND    METHOD. 

a;3_    xy-^f=^-o,     (3)  =  (1)  -  (2). 
x^  +  2:r?/  +3/^  =  9;     (4)  =  (2)^ 


xy  =  2,     (5)  =  (4)  -  (3)  -  3. 
X  —  7/   =  1,      Vide  §  183.  i 

Hence  x  =  2  and  y  =  1. 

In  the  same  way  solve  and  verify  tlie  following  equations. 

(2.)  (3.)  (4.)  (5.) 

(x'-f-/=133.)      (a;3+/=217.)      (:fc^+ 7/^=520.)      {x^-\-7f=nO.) 

(x+y=7.     )      {x+y=l.     )      (a:  H  y  =10.  )      (:c +7/ =iO.  )  \ 

Answers 

a:  =  2,  ^  =  5.      x  =  6,  y  =  l.     a:;  =  8,  ?/  =  2.     a:  =  9,  y=sl 


and       a:    +  y 


^  to  tind  X  and  ?/. 
=  3  (2),  J 


194.  2.  Given  a;^  +  ^  =  9  (1), 

y   =  3  (2), 

THIRD    METHOD. 

x^  J^y^=.{x+  yy  -  ^xy  (x  +  y),  (3).      Vide  §  V4,  ex.  1, 

and  §  132  (2). 
...        (^  4.  y)3  _  3:^y  (a;  +  y)  =  9,  (4). 
That  is,  27  —  9a:y  =  9,  (5)  since  x  -\-  y  =  Z. 

xy  =  2,  (6). 
x-y  =  l,  (7).         §183. 
Hence  x  =  2,  and  y  =  1. 

In  the  same  way  solve   the    equations  of  §  192,  and  §   193, 
and  also — 

(2.)  (3.)  (4.)  (5.) 

(a.-»-y  =  61.)    (^»-3/»  =  342.)    (a;'-y  =  485.)  (.t3-/  =  7.) 

(a:  -3/  =1.    )    (.r   -^^   =6.      )    (.r-y  =5.      )    (.r   -y  =  1.) 

IS 


^GG 


EQUATIONS     "WITH 


195.  1.  Given  a^  ^  f  =  d  (1), 
and       •'»    +  y 


y  to  nnd  X  and  ?/. 

3  (2),  j 


FOURTH    METHOD. 

X  =  o  —  1/.  (3)  =  (2)  transposed. 

a:3  =  27  —  27j/  +  9f  —  f  (4)  =  (3)2. 

.-.     27  —  27?/  +  9j/2  —  ?/^  +  ?/  =  9.  (5)  by  substitution. 
Hence                          x  =  2,  and  y  ==  1. 

In  the  same  way  solve  the  equations — 

(2.)  (3.)  (4.)  (5.) 

(a;3  H-  /  =  35.)  Gx^  +  r  =  ^1-)  C-^'  +f  =  341.)  (a;»  —  ?/'  =  19.) 

(X  H-^  =5.  )  (x  +y  =7.  )  (X  +?/  =11.  )  (X  -^  =1.  ) 


^96.  1.  Given  x^  -\-  f  =  9  (1) 

and       x    -f  y   =  3  (2) 


^'  I  to  fin 

2)  J 


d  X  and  y. 


FIFTH    METHOD. 

Let     X  ==  a  -{-  h,  and  y  =.  a  —  h. 

Then  x  -f  ?/  =  2a  =  3,  and  a  =  ^.     By  addition  and  (2). 
x'  ==  (a  +  Z.)3  =  a^  +  3«2Z>  4-  3«Z;'^  +  Z/^ 
,f  =  (a  _  Z>)3  =  a?  —  3«2^  +  3aZ>2  —  Z.3. 
.x^  -j-  y  =  Id^  -f  6«Z>2  =  9.  By  addition. 

Y  +  9^2  =  9.  Since  «  =  |. 

Hence  h  =  \. 

But  .r.  =  a  +  5  =  I  +  ^  =  2,  and  ?/  =  a  —  ^^  =  -J  —  J  ==  1. 

In  the  same  way  solve  the  equations  in  the  preceding  sections, 
and  also 

(2.)  (3.)  ,     (4.)  (5.) 

Cx»+y=28.)      (a;3— y=26.)      (.x^— y=7000.)      (x3+y=9000.) 

(x  -fy  =4.  )      (X  -y  =2.  )      (x  -y  =10.     )      (x  ^y  =30.     ) 


19^.  1.   Given  :i^  +  y  =  a  (1), 
and       .r    -}-  3^    =  h  (2), 


to  find  X  and  y. 


TWO     U  X  K  X  0  W  X     Q  U  A  X  T  I  T  I  E  S.  207 


-    ^•J/  +  /  =  r,     (3)  =  (1) -^  (2). 


I 


0}  +  2.ry  +  ;?/^  =  ^'^    (4)  =  (2)^ 


l^  — 


a:y  =  -^'     (5)  =  (-i)  -  (o)  ^  3. 


a:_y  =  ±^-^ 


Hence       :.  =  ^^^  ±  \— SX"  )'  '^^ '^  =  i(^  -  V-W  j" 

Vide  §   28,  ex.  19. 

2.  Given  x^  —  f  =  a,) 

V  to  find  X  and  ?/. 
and       a:  —  i/  =  h,  ) 

Apply  tliese  formulas  to  all  tlie  equations  in  §  192  and  §  196, 

inclusiYe, 

19S.  1.  Given  x*  -f  t/*  =  17  (1),  )   ,     .    ,  , 

V  to  find  X  and  i/. 
and       X   +  7/   =  o    (2),  ) 

:,*  +  3/^  =  C:c  +  y)^  -  4^y  (x  +  i/y  +  2x^y^  =  17.     (3) 

Vide  §  132  (2). 
81  —  o6x7/  +  2xy  =  17.     (4)  Since  x  +  3/  =  3. 
Hence  a:^/  =  2,  or  16.     (0)  =  (4^  reduced. 

Then  x  —  7/  =  1,  ot  V  —  55. 

.-.   X  =  2  or  i  (3  +  1/—  55),  and  ?/  =  1  or  ^  (3  —  y^—  55). 
In  the  same  way  solve  the  equations — ■ 

(2.)  (3.)  (4.)  (5.) 

(x*+y=82.)      (x*+3/^=626.)      (x*+7/*=1297.)      (x^+^'^=2Vg-) 
(X  4-y  =4.  )     (:^  +y  =6.    )     (x  +3/  =T.      )     C^  +y  =f    ) 
6.  Given  x*  +  ?/*  =  cr,  and  x  +  ?/  =  Z>,  to  find  x  and  y. 

X  =  i  (6  db  t/- 3Z.-^=|=2T/2a  +  25*). 

J.ns.  "" 

y  =  A  (h  =F  ■/-  362  _p  2  T/2a  +  26*). 

Apply  these  formulas  to  the  above  examples,    (  Vide  28,  ex.  24."^ 


208  E  Q  U  A  T  I  0  N  S     W  I  T  II 


iS9.  1.  Given  x^  —  /  =  992  (1), 

and       X   —  7/   =2      (2), 


I  to  find  X  and  ^. 


0:5  —  y  =  (^^  _  yj  _^  5^^^.  ^:c  _  y-y  _{_  5^^2y  (^^  —  y)  =  992. 

(3)  Tide  §  132  (2). 
32  -f  40.Ty  +  lO^y  =  992.     (4)  Since  :r  —  y  =  2. 
Then  xy  =  8,  or  —  12. 


Hence       a;  =  4  or  —  3,  or  1  rt  y  —  11,  and  ?/  =  3  or  —  4,  or 
—  1  rt  V—  11.      Fif^e  §  16Y,  ex.  17. 

In  the  same  way  solve  the  equations — 

(2.)  (3.)      .  (4.)  (5.) 

(,xH/=33.)      (:^'— /=7/^.)      (a;5-fy5^1056.)     (x5— /=781.) 
(2^  +y  =3.   )       (:c  —y  =1.      )       (x  +y  =6.        )      {x  —y  =1.      ) 

6.  Given  x^  -\-  y^  =  a,  and  x  -\-  y  =■  h,  to  find  a;  and  y. 


200.  1.  Given  x^  J^  7/  =  ^  (1), 

>   CO  unu  ; 

8 


and  xy   =  2  (2), 


y  to  find  X  and  y. 


From  (2)  x^y^  =  8,  hence  x^  =  —3. 

o 
Then  -  -^  y^  =.  9. 

yZ 

Hence  .t  =  2  or  1,  and  ?/  =  1  or  2. 

In  the  same  way  solve  the  equations — 

(2.)  (3.)                        (4.)                      (5.) 

(x'+2/'=351.)  (x*-/=240.)      (rz;5 +3/5=  1267.)  (x'+y'=\l.) 

(      :ry=14.  )  (      xy  =8.     )      (      xy  =12.     )  (      xy  =2.  ) 

^??  sixers. 

x  =  2or7.     x  =  ±4ordb2  \/  ^  1.     x  =  4  or  3.     x  =  2  or  1. 

y  =  7  or  2.     y  =  =b  2  or  ±  2  {/—  1.     y  =  3  or  4.     y  =  1  or  2. 


[■  to  find  X  and  y. 


T  "\V  0     U  N  K  N  0  \S  X     Q  U  A  K  T  I  T  I  E  S.  209 

2©1.  In  an  equation  in  whicli  the  terms  are  liomcgeneous, 
we  may,  with  great  advantage,  introduce  an  auxiliary  unknown 
quantity,  ?jy  letting  x  =  my.  The  value  of  m  can  easily  be 
found,  and  from  this  x  and  y.     Thus, 

1.  Given  X  (x  +  ?/)  =  24  (1), 
and       y  {:c  —  ?/)  =  4    (2), 

These  equations  may  be  written  thus,  by  multiplying  (2)  by  6 : 

x^  -f-  xy  =  24;  and  ^xy  —  6y^  =  24. 
o:"^  -\-  xy  =■  ^xy  —  6j,/^,  or  "x-  =  bxy  —  6y'. 
Now   let  :*"   -=   wy,  whence  x-  =  m-y-^  and  we  have,  by  substi- 
tution in   the  last  equation, 

nry"^  =  bmy"  —  G^^. 

Divide   this   by  y'\  and  vre  have 

r:i^  —  bin  =  —  6. 
Hence  in  =  3  or  2. 

X  =  St/  or  x  =  1y. 
Substitute  the  lust  value  in  (1),  and  we  have 

4/  +  2y  =  24. 
Hence  y  =  =h  2,  and  x  =  zt  4. 

Substitute  the  first  value  of  x  in  (1),  and  we  have 

Hence  y  =  dsz.  l/2,  and  x  =  =i=  o  V' 2. 

In    the   same   way   solve   the   equations — 

2.  X  (x  -f  ^)  =  77,  and  y  (^x  ■-  y)  =  12. 

Ans.  X  =  1  or  V  i/2,  ?/  =  4  or  |  l/2. 

3.  j.^y  +  .xy  =  G,  and  :c^  +  j^  =  10. 

Alls,  .r  =  2  or  1,  y  =  1  or  —  3. 

4.  x'^  -f  a,y  =  12,  and  xy  —  2y^  =  1, 

Ans.  .T  =  ±  3  or  I  v^G,  y  =  ±  1  or  J  |/6. 

5.  ^.?^/2  _|_  y  ^  5^  j^^(j  .^.4  ^  .^-zy  ^  or)_ 


^ns.  X  =  ±  2  or  =b  2  i/—  1,  ^'  =  d=  1  or  ±  V  —  1. 
13 


210  E  (-i  U  ATI  0  N  S     vV  I  T  H 

6    x^u"^  -\-  ?z'  =  20^*,  and  x~  -f-  }/'  =  'IS. 

Ans.   a:  =  rfc  6  or  rp  ^rf ,  y  =  ±  3  or  rh  ^  V'^ —  5. 
7.  x^  -\-  if'  =.  61,  and  a;^  —  xy  =  6.  J.n.s.  .t  =  6,  ^  =  5. 

202.  Sometimes  we  may  introduce  tiro  auxiliary  unknown 
quantities,  one  of  wliicli  represents  the  sum,  and  the  other  the 
product^  of  x  and  y.     Thus, 

1.  Given  x^y  +  xy^  =  6  (1),  and  x^  +  l/^  =  9  (2),  to  find  x  and  y 
These  equations  may  easily  he  written  as  follows : — 

^2/  (.^  +  y)  =  6  (3),  and  (x  +  yf  ~  ^xy  (x  +  y)  =  0,  (4). 
Now  let  x  -\-  y  =  a,  and  xy  =  5,  and  the  equations  become 

ah  =  6  (5),  and  a^  —  3ah  =  9,  (G). 

d^  =  27  or  a=  3,  and  h  =  2. 
Hence  a;  -|-  ^  =  3,  and  xy  =  2,  from  which  w^e  have 

X  =  2,  and    7/  =  1. 

2.  Given  x-*  +  J/^  —  -^y  =  7,  and  x-"*  ~\-  y^  =  35,  to  find  c?-  and  y. 

Ans.  X  =  3,  and  7/  =  2. 

3.  Given  x^  -\-  y''-  -\-  xy  =  28,  and  x^  —  ^^  =  56,  to  find  x  and  y. 

J[?«.s.  X  =  4,  and  y  =  2. 

4.  Given  a;^  +  3/^  _  2^^-^  —  2^7,/^  +    ^^  "^     =15^,  and  ^'^  -f-  / 

x-  -j-  ^ 

=  244,  to    find  x  and  y.     Ail    the    values  of  x    and  y  in    these 
equations  are  as  follows — 

.T  =  8,  and  y  =  1.    x  =  2  zh  3  V^^^,  and  y  =  2  ^  3  V^^l. 
a:  =  1,  and  ?/  =  3. 

/ 


a:==2  (-  1  ±  V  -(1  +  5^0  l/l5),  and  7/=  2  (-lq=  i/_(l  +  ,.i_|/l5.) 
^T=2  (- 1  ±  i/-(l_^L  1/15),  andy=  2  (_l=p i/_(l -,1^1/157) 

203.  Sometimes  it  is  of  much  advantage  to  introduce  two 
auxiliary  quantities,  one  of  which  represents  the  sum,  and  the 
other  the  difference,  of  x  and  y.  Thus, 

1.  Given  x^  —  y^  —  xy"^  -\-  x^y  ■==  25  (1),  and  x^  +  V^  —  '^IJ^ 
~  xhj  =  5  (2). 


THREE     U  X  K  N  0  V,'  X     QUANTITIES.  211 

These  equations  are  easily  transformed  into 

Qvide  §  132  (2),  ex.  7) 
(X  -  ^)  (X  +  ff  =  25  (3),  and  (.x  +  y)  Qx  -  yj  =  5  (4). 
Now,  let  X  +  3/  =  cf,  and  x  —  y  =  h.      By  the  substitution  of 
these  values  in  (3)  and  (4),  we  have 

a^h  =  25,  and  ah''  =  5. 
By  the  multiplication  of  which,  we  have 

a^J/  =  125,  or  ah  =  5.     • 
Hence  a  =  5,  and  h  =.  \. 

X  -f  y  =  5,  and  x  —  ?/  =  1. 
From  which  x  =  3,  and  y  ^=  2. 

2.  Given  x^  +  ^x""  (?/  —  1)  +  3^^  (a:  -f-  1)  +  yS  ^  80,  and 
ic'  -}-  rr  (2y  -f  3)  =  IG  —  y  (y  -\-  3),  to  find  a:;  and  y. 

Ans.  a:  =  4,  y  ==  1. 

3.  Given  x*  —  ?/*  4-  2x^j/  —  2.t^  =  27,  and  x^  —  y^  =  3,  to 
find  X  and  y.  A71S.  x  =  dt=:  2,  y  =  zh  1. 

204.  If  the  preceding  sections,  commencing  at  §  182,  have 
been  studied  with  sufiicient  care,  the  student  will  easily  overcome 
all  the  difficulties  attending  equations  of  the  kind  we  have  been 
examining.     We  will  finish  this  subject  by  adding  a  few 

EQUATIONS    CONTAINING   THREE    Ui^'KNOWN    QUANTITIES. 

1.  Given  X    +  y   +  z   -=6     (1),  >v 

^^  +  y'  +  2;'  =  14  (2),  y  to  find  X,  y,  and  z. 
and  x^  +  z-  =  10  (3),  J 

Ans,  X  =  1,  y  =  2,  z  =1  3. 

2.  Given  Ax^  -f  4/  =  2z'  +  2a'-  (1),  >v 

4^2  _|-  4^2  =  2^2  +  2h^  (2),  I  to  find  x,  1/,  and  2;, 
Ax^  +  4^2  ==  2?/2  4-  2c2  (3),  J 
^«s.  .T  =  ±  A  l/2a-^  +  2c2  —  b',  ?/  =  db  ^  1/252  4-  2a^  —  c^ 


£  =  ±  J-  l/2c^  +  21'  —  a\ 


212  PROBLEMS    IN  VOL  TING 

3.  Given  xy  -f  ^  =  5  (0;") 

xijz  +  s'  =  15  (2),  y  to  find  cc,  y^  and  z. 

xy^  -f  a:2y  —  2x  +  2^  =  8  (3),  J 

J.?is.  X  =  2^  ^  =  1^  2;  =  3. 

PEOBLEMS. 

INVOLVING   TWO    UNKNOWN   QUANTITIES. 
205.  1.     The  ^m    of    two    numbers    is    100,  and  the  dif- 
ference of  the  square  roots  of  the  numbers  is  2.     What  are  the 
numbers  ? 

Let  x^  =  one,  and  y"^  =  the  other  number. 
Then  xJ  -^  y"^  :=■  100,  and  x  —  y  =z  2.     Hence, 
a;2=  64,  andy'=  36. 
2.  The  property  of  A  and  B  together  amounts  to  $13,000,  and 
each  receives   the    same  income.     But  if  A  should  let  his  money 
at  B's  rate  per  cent.,  his   income  would   be   $360,  while    B's  in- 
come at  A's  rate  per  cent,  would  be  $490.     What  is  the  property 
of  each  ? 

Let  X  =  A's  rate  per  cent.,  and  y  =  B's. 

„.       36000         .,  ^         ^  49000        ^, 

Ihen  =  As  property,  and =  B  s. 

y  X 

Therefore,  by  the  question, 

y  X  f  \    y 

36000x        49000?/ 

and       = ^.  (2^ 

y  X  ^  ^ 

36x  +  49y  =  I2>xy.  (3)  =  (1)  reduced. 

6x  =  7y.  (4)  =  (2)  reduced. 

42y  +  49y  =  -^-.  (5)  :=*  (3)  combined  with  (4). 

?/  =  6,  and  x  =  7. 

n^neo    '-%^  ^   S6000,  and  '^  ^  ?7000. 


TWO    UNKNOWN    QUANTITIES.  213 

8.  The  sum  of  two  numbers  is  2-i,  and  their  product  is  35 
times  their  diflference.     AVhat  are  the  numbers  ? 

Let  X  =  the  greater,  and  y  =  the  less  number. 

Then  x  -\-  y  =■  24,  and  xy  =  35  (x  —  ?/). 
Find    the   value  of  x  in    the   first  equation,  and   substitute  it   in 
the  other.  Ans.  14  and  12. 

4.  A    number   divided    by   the    product   of  its    digits   gives   a 

quotient  of  2^.  ^  If   18    be   added  to   the  number,  the  digits  are 

lOx  +  y 
inverted.    What  is  the  number  ?     The  equations  are  =  2|, 

and  10.r  +  y  +  18  =  10^^  +  x.  Ans.  35. 

5.  The  sum  of  the  digits  of  a  certain  number  is  10,  and  if 
the  product  of  the  digits  be  increased  by  40,  the  sum  is  the 
number  inverted.     What  is  the  number  ?  Ans.  46. 

6.  The  sum  of  two  numbers  is  7-^,  and  the  sum  of  the  third 
powers  is  343^.     Yv'hat  are  the  numbers  ?  Ans.  7  and  i. 

7.  The  sum  of  two  numbers  is  47,  and  their  product  is  546. 
What  is  the  sum  of  their  squares  ?  Ans.  1117. 

8.  The  sum  of  two  numbers  is  20,  and  the  product  is  99. 
What  is  the  sum  of  their  cubes?     (T7^el3*2  (2).)      Ans.  2060. 

9.  The  sum  of  two  numbers  is  8,  and  the  product  is  15. 
What  is  the  sum  of  their  fourth  powers?  Ans.  706. 

10.  The  sum,  product,  and  difference  of  the  squares  of  two 
numbers  are  all  equal.     What   are  the  numbers? 

Ans.  i  (3  ±  y/5)  and  A  (1  ±  |/5). 

11.  The  sum  of  the  squares,  the  product  of  the  squares,  and 
the  difi'erence  of  the  fourth  powers  of  two  numbers  are  all  equal. 
What  are  the  numbers?  Ans.  1.27203,  and  1.61808. 

12.  The  sum  of  the  fourth  powers  of  two  numbers  is  a,  and 
the  product  h.     What  are  the  numbers  ? 

Ans.   V\  (a  ±  j/cr  -  46^  V\  (a  =F  \/a^  —  4i* 


214  PROBLEMS    INVOLVING 

13.  Divide  GO  into  two  parts  so  that  the  product  of  the  parts 
shall  be  to  the  difference  of  their  squares  as  2  to  3. 

Ans.  40  and  20. 

14.  There  are  two  numbers  whose  product  is  77,  and  the 
difference  of  their  squares  is  to  the  square  of  the  difference  as 
9  to  2.     What  are  the  numbers  ?  Ans.  11  and  7. 

15.  The  product  of  two  numbers  is  48,  and  the  difference  of 
their  cubes  is  to  the  cube  of  the  difference  as  37  to  1.  What 
are  the  numbers?  ^l^is^S  and  6. 

IG.  The  difference  of  the  fourth  powers  of  two  numbers  divided 
by  the  difference  of  the  numbers  is  2336,  and  the  product  of  the 
difference  of  their  squares  by  the  difference  of  the  numbers  is 
576.     What  are  the  numbers  ?  Ans.  11  and  5. 

17.  The  product  of  two  numbers  is  320,  and  the  difference  of 
their  cubes  is  equal  to  61  times  the  cube  of  their  difference. 
AVhat  are  the  numbers  ?  Ans.  20  and  16. 

18.  Divide  a  number  a  into  two  parts,  so  that  the  greater  part 
may  be  a  mean  proportional  between  the  whole  number  and  the 
less  part.     Let  x  =  the  greater  part,  and  ^  =  the  less. 

Then  X  -{-  7/  =  a,  and  a  :  x  :  :  x  :  t/. 

Ans.  I  (3  -  1/5),  and  ^  (~  1  +  V^.) 

lit  Li 

If  a  =  20,  then  the  numbers  are  12.36  and  7.64. 

19.  The  sum  of  two  numbers  is  a,  and  the  sum  of  their  re- 
ciprocals is  h.     What  are  the  numbers  ? 

,        a  \a}        a        ,  a  \a?        a 

^«-  2  +  Vi  -  J'  ^""^  2  -  Vl  -  %■ 

20.  The  sum  of  the  squares  of  two  numbers  is  a,  and  the 
sum  of  the  reciprocals  of  the  numbers  is  h.  What  are  the  sum 
and  product  of  the  numbers  ? 

1  , 1  / 

Ans.  Sum,  -  (1  ±  l/l  +  ah')',  product,  ^  (1  ±  T/l  +  al?). 

If  a  =  5  and  5  =  l^i,  then  the  numbers  are  1  and  2. 


TWO     UNKNOWN     QUANTITIES.  215  , 

I 

21.  A    mercliivut    bought    51    gallons  of   j^Iadcira   wine,    aud    a        | 

certain    quantity  of  Teneriffe.     For    the    former   he    gave  half  as 

many  shillings    by  the    gallon  as  there  were  gallons  of  Teneriffe, 

and    for    the    latter  1  shillings    less    by  the    gallon.     He  sold  the 

mixture  at  10  shillings    by  the  gallon,   and   lost  £28  10s.  by  his        I 

bargain.      Required    the    price  of   the    Madeira   and  the  quantity        : 

of  Teneriffe. 

Ans.  Madeira,  18  shillings;  Teneriffe,  36  gallons. 

22.  The   side   of   one    square    garden    exceeds    the    side  of  an- 
other by  5  rods,    and    both    gardens    contain    1025    square    rods.        ■ 
What  is  a  side  of  each  ?  Ans.  20  and  25.        : 

23.  A  farmer  has  a  field  16  rods  long  and  12  rods  wide.  He 
wishes  to  enlarge  the  field  so  that  it  may  contain  twice  as  much 
area,  and  not  change  the  proportion  of  the  sides.  What  will  be 
the  sides  of  the  field?  Ans.  Length,  16  l/2';  breadth,  121/2". 

24.  A  rectangular  grass-plat  has  its  sides  in  the  ratio  of  4  to  .; 
3.  A  walk  outside  the  plat,  6  feet  wide,  contains  ^tt  as  much  [ 
ground  as  the  plat  itself.  AThat  is  the  length  and  breadth  of  , 
the  plat  ?                                 Ans.  Length,  342.72;  breadth,  257.04. 

25.  A  grocer  sold  80  pounds  of  mace  and  100  pounds  of 
cloves  for  £66,  and  finds  that  he  has  sold  60  pounds  more  of 
cloves  for  £20  than  of  mace  for  £10.  What  was  the  price  of 
each  per  pound  ?  Ans.  10  shillings  and  5  shillings. 

26.  Find  two  numbers  whose  sum  multiplied  by  the  second  i 
is  84,  and  whose  difference  multiplied  by  the  first  is  16. 

Ans.  d=  8  and  db  6,  or  zp  l/2  and  ±  7  "l/2 

27.  The  square  of  the  sum  of  the  squares  of  two  numbers  is 
169,  and  the  product  of  the  squares  is  36.  What  are  the  num- 
bers ?  Ans.  =h  3  and  ±  2,  or  ±  3  V^^  and  ±  2  V^^^ 

28.  The  difference  of  the  fourth  powers  of  two  numbers,  mul- 
tiplied by  the  product  of  the  squares,  is  147,600.     The   sum  of         ! 


216  PROBLEMS     INVOLVING 

tlie    squares    multiplied   by  the   product  of   the   numbers    is   820 
AYhat  are  the  numbers  ? 

Ans.  db  5  and  ±  4,  or  db  5  V  —  1  and  it  4  l/—  1. 
29.  A  and  B  bought   a   farm    containing  a  acres,  each  paying 
m  dollars.     A  paid  h  dollai"S  per  acre  more  than  B,  in  considera- 
tion of  taking  his  share  from  the  best  portion.     What  does  each 
one  take,  and  at  what  price  per  acre  ? 

Ans.  A  takes  ^''''  at  2m  +  c.^  +  l/4m- +  ^^^l 

2m  -\-ab  -{-  V'^m''  -\-  a'U'  2a 

B  takes  g^^^!i-  at  2m-a5  +  T/4m-  +  a-^-. 

2m  —  ah  +  l/4m^  -f  a'b''  2a 

Tc                ^1,       A  *  1                     2a                     2  4-  h  +  l/4  +  b^ 
If  j)i  =  a,  then  A  takes ^  at      ~      ~    — , 

2  +  i  +  l/4  -f  6^  2 

and  J3  takes r:=r  at - ' > 

2  _  ^,  -j-  v  4  +  ^'  2 

In  this  case — that  is,  when  each  pays  as  many  dollars  as  there 
are  acres — the  price  per  acre  does  not  depend  upon  the  number 
of  acres  purchased. 

If  m  =  a  =  200  and  &  =  |  or  $.75,  then  A  takes  81.867  acres 
at  S2.443,  and  B  takes  118.133  acres  at  $1,693. 

If  m  =  a  =  200  and  b  =  $2,  then  A  takes  58.579  at  $3.41421 
per  acre,  and  B  takes  141.421  at  $1.41421  per  acre. 

If  m  =  a  =  300  and  b  =  $1.50,  then  A  takes  100  acres  at 
$3.00  per  acre,  and  B  takes  200  acres  at  $1.50  per  acre. 

206.  Every  equation  must  be  regarded  as  the  algebraic  con- 
ditions of  some  problem.  If,  therefore,  on  solving  the  equation 
the  value  of  x  is  imaginary,  it  is  absolutely  impossible  to  fulfil 
the  conditions  of  the  problem. 

80.  Divide  10  into  two  parts,  so  that  the  product  shall  be  26. 

Let     X  =  one,  and  1/  =  the  other  part. 

Then  X  4-  y  ==  10,  and  X2/  =  26. 

Hence  x—  r  --  2  ]/—  1,  and  a;  =:^  5  4-  i/—  1, 7;  =  5  —  l/—  1 


TWO    UNKNOWN    QUANTITIES.  217 

Therefore   it   is   impossible   to  divide    10    into    two   parts  so  that 
the  product  shall  be  26.     This  is  readily  seen  on  trial,  thus  : 

10  =  9  +  1,  and  9  X  1  =  9. 

10  =  8  +  2,  and  8  X  2  =  16. 

10  =  7  +  3,  and  7  X  3  =  21. 

10  =  6  +  4,  and  6  X  4  =  24. 

10  =  5  +  5,  and  5  X  5  =  25. 

We  see   that   the  product  is   greatest   when  the  pai-ts   are  equal. 
That  this  is  generally  the  ease  may  readily  be  shown. 

Let  X  =  one  part,  y  the  other,  2s  the  sum,  and  2d  the  diflference. 

Then  x  -\-  y  =  2s  and  x  —  y  =■  2d. 

Whence  x  =  s   +  c?"|     (1). 

and  ^  ==  s   —  d )     (2). 

The  product  of  which  is  xy  =  s^  —  d^.     (3). 

It  is  plain  that  the  second  member  of  (3)  increases  as  d 
diminishes,  and  therefore  that  it  is  the  greatest  when  d  =  0,  i.e. 
when  there  is  no  difference  between  the  parts. 

31.  Divide  20  into  two  parts,  so  that  the  product  shall  be  150. 

Ans.  a:  =  10  +  5  V^^^,  ?/  =  10  —  5  V^^- 

32.  The  sum  of  two  numbers  is  1,  and  the  sum  of  their  re- 
ciprocals 2.     What  are  the  numbers?  (^Vide  IG'T,  ex.  18.) 

Ans.  i(l  -{-  V^^),  and  ^  (I  —  V"^^^). 

33.  If  4  is  added  to  a  certain  number,  and  the  sum  is  divided 
by  the  number  itself,  the  quotient  is  the  same  as  that  obtained 
by  dividing  three  times  the  number  by  the  number  diminished 
by  4.     What  is  the  number?  Ans.  ±  v  —  2. 

19 


218       EQUATIONS     OF    THE     SECOND    DEGREE. 


GENERAL    PROPERTIES    OF    EQUATIONS    OF    THE    SECOND 

DEGREE. 

Dejinitlons. 
20'?'.  1.  A  root  of  an  equation  is  a  quantity  which,  being  sub- 
stituted for  X  in  the  given  equation,  satisfies  it. 

2.  An  imaginary  root  is  one  involving  the  expression  y —  1. 

3.  A  real  root  is  one  not  involving  an  imaginary  quantity. 

4.  Equal  roots  are  where  the  roots  are  the  same  quantity, 

EQUATIONS    OP   THE    SECOND    DEGREE   HAVE   TWO    ROOTS,  AND 

ONLY   TWO. 

Demonstration. 

20S.  Every  equation  of  the  second  degree  can  be  reduced  to 

the  form  of 

x^  -{-  2/^-^  =  9.' 
Add  'p^  to  both  members,  and  we  have 

x^  4"  2px  ~{-  p^  =  p^  -\-  q^ 

or  {x  -\-  jyy  =2^  +  Q> 

and,  by  transposition, 

(X  +  py  -  (p2  +  2)  =  0, 

or     (x  +  p  +  Vp^~+~q)  (x  -{-p  —  y'pn^Tg^  =  0. 

(Vide  "YT.) 
Divide  this  equation  first  by  one  factor,  and  then  by  the  other, 
and  wc  have 

33  4-  p  _  >/^7T7=  0  .-.  X  =  ~^9  +  l/pM^  (1)^ 
j.nd  [q.E.]). 

X  —  p  -{■  V'f  +  q  =  0   .-.   X  =  —  V  —  Vp^  +  q.    (2)  ' 

209.  If  J)  =  0,  then  x  =  \/q,  and  x  =:  —  \/q^  which  are  the 
roots  of  the  incomplete  equation  a.'^  -f  2  x  0.  x  =  q,  or  x^  =  q. 
(  Vide  158  (3).) 


EQUATIONS     OF    THE     SECOND     DEGREE.       219 

210.    The  aJQehraic  sum  of  the    fico    roofs  is    equal   to   the  co- 
efficient of  the    second    term  icith    its    sign    changed  ',   for 


The  roots  are  .r  =  —  p  -f  V i^^  -f  5',  and  x  =  —  jf  —  l/p''  +  q^ 
the  sum  of  Avhich  is  —  2j?. 

211.  77ie    product    of    the    two    roots    is    equal   to    the    second 
term    icith    its    sign    changed ;    for 

The  roots  are  x  =  —  ^)  -]-  |/j>^  -j-  q,  and  x  =  —  p  —  \/p^  -f  q^ 
the  product  of  which  is  —  q. 

APPLICATION    OE    THESE    PROPERTIES. 

212.  1.  What    is  the  equation  whose  roots-  are  5  and  —  9  ? 

By  210.     2p  =  4,  and  by  211^  q  =  45. 
Therefore  the  equation  is     x^  +  4x  =  45. 

2.  What  is  the  equation  whose  roots  are  4  and  1  ? 

Ans.  x^  —  ^x  =  —  4. 

3.  What  is  the  equation  whose  roots  are  9  and   —  1  ? 

Ans.  x^  —  Sx  ==  9. 

4.  What  is  the  equation  whose  roots  are  —  2  and   —  2  ? 

Ans.  x^  -f  4x  =:  —  4. 

5.  What  is  the  equation  whose  roots  are  1  and   —  1^^? 

Ans.  x'^  4-  /-^  =  1^^,. 

6.  What  is  the  equation  whose  roots  are  7  and   —  8  ? 

Ans.  x^  -\-  X  =  56. 

7.  What  is  the  equation  whose  roots  are   —  7  and  —  7  ? 

Ans.  x'  4-  14x  =  —  49. 

8.  What  is  the  equation  whose  roots  are  7  and  7? 

Ans.  x^  —  14:x  ==  —  49. 

9.  What  is  the  equation  whose  roots  are  a  and  h  ? 

Ans.  x^  —  (^a  -\-  h)  X  =  —  ah, 


220      EQUATIONS     OF     THE     SECOND     DEGREE. 

-1^    iTTi    ,     •     .1  •  1  a\/b  4-  hi/a         , 

10.   What    IS  the  equation  whose   roots   are  ; ,  and 

a  —  6 

a-[/b  —  h\/a  2al/l>x        —  ab 

■ J .  Ans.  x^ —  =  7' 

a  —  0  a  —  b         a  —  6 

213.  Advantage  may  be  taken  of  these   general   properties  in 
solving  any  equation  of  the  second  degree. 

1.  Given  x^  —  Sx  =  —  15,  to  find  the  two  roots. 

Let  X  =  one  root,  and  i/  =  the  other. 

Then  x  +  ?/  =  8,  By  210. 

and  xy  =  15.  By  211. 

AYhence  x  =  ^,  and  ^  =  3.  Vide  182,  ex.  1. 

2.  Griven  a;^  -f  6.c  =  187,  to  find  the  two  roots. 

Here  a;  -}-  y  =  —  6, 

and  ^y  =  —  187. 

Hence  x  =  11,  and  i/  =  —  17. 

3.  Griven  x^  +  4x  =  —  4,  to  find  the  roots. 

Here  x  -{-  ?/  ==  —  4, 

and  x?/  =  4. 

Hence  x  =  —  2,  and  y  =  —  2. 

7x 


4.  Given  Sx^ 

+ 

1  .^ 

5 

=  4|,  to  find  the  roots. 

By  reductio 

n 

rf.2       .         "^-^    _     2,2 

^    +   15  -  ^^• 

Then 

^    +  ^  =  -  t\, 

and 

^!/  ==  —  f  f  • 

Hence 

X  =  1,  and  y  —  —  \J^ 

5.  Find  the  roots  of  ox'^  _  2x  =  8.     (^Y'ule  1^76,  ex.  4.) 


Ans.  2  and  —  1^. 


6.  Find  the  roots  of  — ;; j ;— ^  =  5|.         Ans.  4  and  1. 


X  -\-^  7x 

~x        ^  X  -f  8 
7.  Find  the  roots  of  x""  —  20:/;  =  —  104. 

Am.  10  +  2]/~in  and  10  —  2l/^^ 


EQUATIONS     OF     THE     SECOND     DEGREE.      221 

8.  Find  the  roots  of  cc^  —  (a  —  V)  x  =  a.      Ans.  a  and  —  1. 

9.  Find  the  roots  of 6  =  — —-- 

X  —  a  a  -\-  X 

Ans.  (  Vide  ll'O,  ex.  6.) 

10.  Find  the  roots  of  x''  —  IOjc  =  —  26. 

Vide  li^G,  ex.  44,  and  206,  ex.  30. 

314.  Every  equation  of  the  second  degree,  as  we  have  stated, 
may  be  reduced  to  the  form  x"^  -f-  2px  =  q  (1),  the  signs  not 
being  considered. 

This  form  is  obtained  when  p  and  q  are  hoth  j)ositive. 

If  p  is  negative  and  q  positive^  we  have 

x^  —  2px  =  q  (2) 

If  p  is  p)ositive  and  q  negative,  we  have 

x'  +  2i)x  =  —  q  (3) 

If  p  is  negative  and  q  negative^  we  have 

x^  —  2px  =  —  q  (4) 

And  these  are  all  the  combinations  that  can  be  made  in  the 
signs;  for  if  x"^  is  negative,  we  may  multiply,  or  divide,  the 
whole  equation  by  —  1,  and  it  will  be  found  in  one  of  the  above 
forms.     (Vide  1^6,  5.) 

The  roots  of  these  equations  are  respectively  (^ide  IT'l,  l'y4) 


X  =  —  p  d=  Vp'  +  q.  (1) 

x=:         p±  Vp'  +  q.  (2) 

X  ■=  — p  dtz  V'p'^  —  q.  (3) 

X  =        2>  ^  V p"^  —  q-  (4) 

From  these  roots  we  easily  deduce  the  following  facts: — 

1.  The  roots  of   (1)  and  (2)  are  always  real. 

2.  |/p2  -f  ^  >  p.    .-.  The  first  root  of  (1)  and  (2)  is  positive^ 
the  second  negative. 

3.  The  negative  root  of  (1)  is  numerically  the  larger. 


222  RATIO     AND     PROPORTION. 

4.  The  positive  root  of  (2)  is  numerically  the  larger. 

5.  If  5"  <^  jj^i  the  roots  of   (3)  and  (4)  are  hotli  real. 

6.  If  2'  <C  i^^  l^oth    roots  of   (3^    are    negative,  and   both  roots 
of  (4)  are  positive. 

7    If  2'  ==  P^  ^^G  roots  of   (3)  and  (4)  are  equal. 

8.  If  q  ]>p^,  the  roots  of  (3)  and  (4)  are  imaginary.     {Yide 

9.  If  p  =  0,  the  roots  of  (1)  and  (2)  are   numerically  equal. 
but  of  contrary  signs. 

10.  If  p  =  0,  the    roots  of   (3)    and    (4)    are    imaginary    and 
equal,  but  of  contrary  signs. 

11.  If  2'  =  0,  the  first   root  of  (1)    and    (3)  is  0,  the   second 
-2p. 

12.  If  2'  =  0^  the  first  root  of  (2)  and  (4)  is  2j),  the  second  0. 

13.  If  2?  =  0;  and  ^  =  0,  the  roots  of  (1),  (2),  (3),  and  (4), 
are  all  0. 

RATIO   AND   TROPORTION. 

215.  Ratio  is  the  quotient  which    is    obtained    hy  dividing  one 
quantity  hy  another  of  the  same  hind.     Thus, 

The  ratio  of  a  to  Z>  is  -,  commonly  expressed  by  a  :  h. 

1.  The  two  quantities  forming  a  ratio  are  together  called  terms. 

2.  The  first  term  alone  is  called  the  antecedent. 

3.  The  second  term  alone  is  called  the  consequent.     Thus, 

a  :  h  are  the  terms,  a  is  the  antecedent,  and  h  the  consequent. 

216.  A  proportion  is  an  equality  of  ratios.     Thus, 

-  =  -,  commonly  written  a  :  h  : :  c  :  d. 
1.  The  first  and  last  terms  of  a  proportion  arc  called  extremes. 


RATIO     AND     PROPORTION.  223 

2.  The  second  and  third  terms  are  called  means.  I 

i 
8.  Tlie  first  and  second  terms  form  tlic  first  coujilet. 

4.  The  third  and  fourth  terms  form  the  second  couplet.  \ 

5.  The  last  term  is  a  fourth  proportional  to  the  other  three. 
Thus,  I 

a  and  d  are  extremes^  h  and  c  are  means,  a  and  h  the  first  i 

couplet,  and  h  and  d  the  second  couplet. 

I 

6.  If   the  means  of  a    proportion    are    the    same  quantitjj,  that        j 

quantity  is    called  a  mean  proportional   between    the    other    two  j        ' 
and    the    last  term  is  a  third  proportional   to    the  first  term  and 
one  of  the  means.     Thus,  in  the  proportion 

a  \h  \'.h  \  Cj 

6  is  a  mean  proportional  between  a  and  c.  and  c  is  a  fourth  pro- 
portional to  a  and  h.  I 

7.  A  continued    pronortion    is    one  in  which  several   ratios    are        I 
equal.     Thus,  i 

a  :  b  : :  c  :  d  : :  ni  :  n  :  :  p>  :  q,  &c.  I 

8.  Three  or  four  quantities  are  in  harmonical  jij-ojyorfion  when 
the  first  is  to  the  last,  as  the  difference  between  the  first  two  is 
to  the  difference  between  the  last  two.     Thus, 

a,  h,  c,  are  in  harmonical  proportion  when  a:  c  '.:  a  —  Tj  :h  —  c.        , 
a,  &,c,andf?,      ''  "  ''  ''     a'.d::a  —  h:c  —  d.        \ 

21f .  Quantities  are  in  proportion  b}''  alternation,  when  ante- 
cedent is  compared  with  antecedent  and  consequent  with  con- 
sequent. I 

21S.    Quantities    are    in    proportion    by    inversion,  when    ante-        ; 
cedents    are    made    consequents    and    consequents   are  made   ante- 
cedents. 


224  RATIO    AND     PROPORTION. 

219.  Quantities  are  in  proportion  by  composition,  when  the 
Bum  of  antecedent  and  consequent  is  compared  with  either  ante- 
cedent or  consequent. 

220.  Quantities  are  in  proportion  by  division,  when  the  dif- 
ference of  antecedent  and  consequent  is  compared  with  either 
antecedent  or  consequent. 

221.  Two  varying  quantities  are  reciprocally  or  inversely  pro- 
portional when  one  is  increased  as  many  times  as  the  other  is 
diminished.     Thus, 

V  y  y 

icx^=2xx  -=3xX  o=  ^^^  X  -^  =  xy,  the  product  hQ\n^  fixed. 

Li  O  Ifth 

222.  Equimultiples  of  two  quantities  are  the   results  obtained 

by  multiplying  both  by  the  same  quantity.     Thus, 

ma  and  mh  are  equimultiples  of  a  and  I). 

223.  Proposition  I.  If  four  quantities  are  in  proportion, 
the  product  of  the  extremes  is  equal  to  the  product  of  the  means. 

For,  since  a  :  h  : :  c  :  d,  we  have   -  =  _.  (1) 

ct        o 

Clear  (1)  of  fractions,  and  we  have  ad  =  he.  (2) 

Cor    1.    If  6  =  c,  then  (2)  becomes  ad  =  h"^.  (3) 

That  is.  The  product  of  the  extremes  is  equal  to  the  square  of 

the  means. 

Cor.  2.    If  both  members  of  (2)  be  divided  by  ac,  we  have 

d        h    .      h        d 

-  =  -,  I.e.  -  =  -,  or  a  :  0  ::  c  :  d.         (iZ) 

c        a  a        c 

That  is,  If  the  product  of  two  quantities  he  equal  to  the  pro- 
duct of  two  other  quantities,  the  first  two  may  he  made  the  ex- 
tremes, and  the  second  two  the  means,  of  a  proportion. 


RATIO    AND    PROPORTION.  225 

Prop.  II.     If  four  quantities  are  in  proportion^  they  will  he  in  j 
proportion  hy  alternation. 

For,  since         a  :  6  : :  c  :  cZ,  we  have  ad  ^  he.     (2)     (Prop.  1.)  i 

Dividing  both  members  of  (2)  by  dc^  we  have —  i 

ad         he    .        a        h  z      j        /^n^  I 

-—=.—-,  I.e.  -  =  -,  whence  a  '.  c  w  b  \  d.      (lU)  I 

dc         dc  c        d  I 

Prop.  III.     If  four  quantities  are  in  proportion^  they  will   he        I 
in  proportion  hy  inversion.  \ 

For,  since     a  :  5  : :  c  :  cZ,  we  have  ad  =  he.  or  -—  =  — ;.      (4")  i 

he       ad  -'  ^ 

Multiply  both  members  of  (4)  by  ac,  and  we  have  ' 

ac        ac    .       a        c       ,  ,  y  ^--. 

-—  =  — •,  I.e.  7  =  -»  whence  b  :  a  : :  d  :  c.         (11) 
be        ad  b        d  ^  , 

Prop.  IV.     If  four  quantities  are   in  proportion,  they  will  he 
in  proportion  hy  composition  or  division.  i 

For,  since  a  :  h  : :  c  :  d,  'we  have    ad  =  he.         (2)  I 

Add  or  subtract  hd  according  to  AX.  I.,  and  we  have — 
ad  db  hd  =  he  ztz  hd,  or  («  zh  h)  d  =  (c  zh  d')  h,  whence 
a±:h  :h  ::  c±  d  :  d.     (5)     (Prop.  1,  Cor.  2,  and  Prop.  3.) 
From  (2)  we  also  have    ac  dz  ad  =  ac  db  he,  or  (c  db  (?)  a  =        j 
(a  ±  h)  c,  whence  | 

adzh  :a::cdzd:d.         (6)        (Prop.  1,  Cor.  2). 

Prop.  V.     If  four  quantities  are  in  proportion,  the  sum  of  the        I 
first  and  second  icill  he  to  their  difference,  as  the  sum  of  the  third 
and  fourth  is  to  their  difference. 

For,  by  (5)  and   Prop.  2,  we  have —  \ 

7              7      7      7    .       c  -f  tZ        cZ  1 

a  -\-  b  '.  c  -\-  d  '.'.  b  '.  d,  i.e.  r  =  T-  ' 

c  —  d        d 

and  a  —  b  :  c  —  d  ::  b  :  d,  i.e.  j  =  t^-  I 

'  a  —  h         o 

whence 
c  d-  d  ^  cj-_^,  ^^^^  is,  a  +  Z; :  a  —  Z) : :  c  -j-dic—d.    (7)  (Prop.  2.) 


a 


d-  h         a  —  b 


226  RATIO    AND    PROPORTION. 

Prop.  VI.  Equimultiples  of  two  quantities  are  proportional 
to  the  quantities  themselves. 

For  —  =  -,  or  ma  :  mh  ::  a  :  h.  (8) 

ma        a 

mo         izo 
Cor.       Since       —  =  — ,  we  have  ma  :  mh  wna:  nh.      (9) 
ma        na 

Prop.  VIT.  If  four  quantities  are  in  proportioiij  the  like 
powers  or  like  roots  loill  he  in  prop)ortion. 

For,  since  a  :  h  : :  c  :  d,  we  have  ad  =  he,  or  a'^d'^  =  h'^c"',  in 
which  m  is  a  whole  number  or  a  fraction.  Restore  the  propor- 
tion, and  we  have 

a'"  :  h""  ::  c"^  :  <Z"^.  (13) 

Prop.  VIII.  If  two  sets  of  quantities  are  in  propor^tion^  th6 
products  of  the  corresponding  terms  will  he  in  proportion. 

For,  since         a  :  Z>  : :  c  :  c?,  we  have  ad  =  he; 

and  since  m  :  n  w  p  :  q^  ViQ  have  mq  =  np ; 

whence  am  X  dq  ==  hnXcp>,  or  am  :  hn  : :  cj)  :  dq.      (14) 

Prop.  IX.  In  a.  continued  proportion,  the  sum  of  all  the  ante- 
cedents  is  to  the  sum  of  cdl  the  consequents  as  any  one  ante- 
cedent is  to  its  consequent.     (^Vide  §  2,  def.  7.) 

For,  since     a  '.  h  ::  c  '.  d,  ^e.  have  ad  =  he ; 

and  since      a  :  6  : :  m  :  w,  we  have  an  =  hm; 

and  since  a  :  h  ::  a  :  h,  we  have  ah  =  ah.  The  sum  of  these 
equations  is 

(jj^cl-\-n)  a  =  (a-]-c-^m)  h,  or  a-{-c-\-m  :  h-j-d-{-n  ::  a  :h.       (15) 

Prop.  X.  If  the  first  couplets  of  two  proportions  are  the  same^ 
the  second  couplets  will  form  a  prop>ortion. 

For,  since     a  :  h  : :  c  :  d,  and  a  :  h  ::  e  :f, 

we  have        -  =  — ,  and  —  =  — . 

a         c  a         e 

.'.   —  =  ^—1  whence  c  :  d  ::  e  :  f  (10) 

c  e 


\ 


PROBLEMS. 


227 


Cor.     By  alternation,  If  tlie  antecedents  of  tico  2:)roportions  are 
the  same,  the  consequents  v:ill  he  proportional. 

ILLUSTRATION   OF   THE   PRECEDING  PROPOSITIONS. 
4:3::20:15.     By  Prop.  1st,      4  x  15  =  3  x  20,  i.e.  GO  =  60 

2nd,  4  :  20  : :  3  :  15. 
3d,  3  :  4  : :  15  :  20. 
4t]i,      4  ±  3  :  4  : :  20  ±  15  :  20, 

7  :  4  : :  35  :  20, 


4:3: 

:20:15. 

u 

4:3: 

:  20  :  15. 

il 

4:3: 

:20:15. 

a 

I.e. 


1:4::    5  :  20. 


4:3::20:15 
4:3::  20:  15 


5th,      4+3  :  4-3  ::  20  +  15  :  20-15, 


i.e.  7  :  1  :.35  :  5. 
"         7tli,      4--^  :  3^  : :  20^  :  25^ 

i.e.  16  :  9  ::400  :  225. 
4:3::20:15::44:33.    By  Prop.  0th,  4  +  20+44:3  +  15  +  33::4:3, 

i.e.  68  :  51  ::4  :  3. 


4  :3 

5  :8 
4  :3 
4  :  3 


20  :  15 
10  :  16.  j 
20  :  15, 
44  :  33. 


I     By  Prop.  8th,       20  :  24  : :  200  :  240. 
By  Prop.  10th,     20  :  15  : :  44  :  33. 


PROBLEMS. 

224.    1.    Any    three    terms    of    a    proportion    being    given,   to 
find  the  other  term. 

In  the  proportion  a  :  h  :'.  c  :  cl, 

let  X  take  the  place  of  «,  5,  c,  and  d,  in  succession.     In  each  case 
we  are  to  find  the  value  of  x. 


X  :  h  ::  c  :  d,  whence  dx  =  he,  i.e.  x  =  — •  (1) 

(2) 


a  :  X  ::  c  :  d,        ^'        ex  =  ad,  i.e.  x^ 


a  :  h  ::  X  :  d,        "        hx  =  ad,  i.e.  x  = 


d 

ad 

c 

ad 


(3) 


he 


a  :  h  : :  c  :  Xj        "        ax  =  he,  i.e.  x  —  — •  (4) 


228  PROBLEMS. 

(1)  and    (4)    show    that    either  extreme  is  equal  to  the  j)rod>ict 
of  the  means  divided  hy  the  other  extreme. 

(2)  and    (3)    show  that  cither    mean    is    equal   to    the  product 
of  the  extremes  divided  hy  the  other  mean. 

2.  Find  the  value  of  x  in  the  proportion  x  :  2  : :  b  :  1.     Ans.  10. 

3.  Find  the  vakie  of  x  in  the  proportion  12  :  x  ::  4  :  7.  Ans.  21. 

4.  Find  the  value  of  x  in  the  proportion  8  :  5  : :  x  :  10.  Ans.  16. 

5.  Find  the  value  of  x  in  the  proportion  14  :  12  : :  7  :  x.  Ans.  6. 

6.  To  find  a  mean  proportional  be'twcen  two  quantities. 

In  the  proportion  a  :  x  : :  x  :  cZ,  we  have  x"^  =  ad  .*.  x  ==  y  ad. 
Hence,  the    mean    'proportional    hcticccn    two    quantities    is    the 
square  root  of  the  product  of  the  quantities. 

7.  Find  the  mean  proportional  between  9  and  4.     Ans.  "]/ o6  =  6. 

a  :  X  ::  y  :  h,')    to  find  the   relations 


8.  Given  the  proportions  7    t  p 

a  :  m: : p)  '•  l>:  J        oi  x^  y,  m,  and  p. 

Ans.  X  '.  m  '.'.  p  '.  y. 

9.  Given  x"  :  (14  —  xj  : :  16  :  9,  to  find  x. 

X  :  14  —  X  : :  4  :  3.  Prop.  8th. 

-   X  :  14  :  :  4  :  7.  Prop.  4th. 

7x  =  56.  Prop.  1st. 

X  =  8. 

10.  Given  xy  =  24  and  x^ — y^  :  (x — yy  : :  19  :  1,  to  find  x  and  y. 

We  have 
a;' — y^:x^ — ^x^y-j-Sxy"^ — y^:  :  19  : 1. 

3xV  —  3x3/2 :  (^x  _  ^)3  : :  18  : 1.    Prop.  4. 
xy  (x  —  y)  :  (^  —  3/)^  '•'    6:1.    Dividing  antecedents  by  3. 

xy  :  (x  —  yy  ::    6:1.    Dividing  1st  couplet  by  (x — ?/). 

24  :  (x  —  3/)2  : :  6:1.    Since  xy  =  24. 

4:(x  — ?/)2::  1:1. 

Hence     x  —  y  =  2  and  xy  =  24,  whence  x  =  6  and  y  =  4, 

11.  Given  (a  +  '-^T  •  («  —  ^0^  : :  x  +  y  :  x  —  y,  to  prove  that 

a  :  X  ::  l/2a  —  y  :  ^/y. 


ARITHMETICAL     PROGRESSION.  229 

a^-{-2ax-^x^  :  a^ — 2ax-\-x'^  ::  x-\-y  :  x — i/.     By  expanding. 
2a3  +  2x2  .  4^^  : :  2x  :  2y.  Prop.  5th. 

a^  -\-     x^  :  2ax  : :    x  :  y.  Dividing  by  2. 

Transferrins;  the  fac- 


a?  -{■    X   .  2a     ::  x^  '.  y. 


(  Trt 

I      1 


o 

tor  X. 


a?  -\-    x^  :    x^  \\2a'.y.  Prop.  2nd. 

a^  :  x^  : :  2a  —  y  '•  V-  Prop.  4th. 


a  :  X   : :  l/2a  — ^  :  l/y.  Prop.  7th. 

12.  Giren     xy  =  135   and   x^  —  ^^  :  (x  —-  ?/)2  : :  4  :  1,  to   find 
X  and  y.  ^?is.  x  =  15;  y  =  9. 

(X  —  3/:x  +  ?/::2:3;) 

13.  Given    \  o     r    r  ^^  ^"'^  ^  ^^^  V- 

(  X  +  ?/  :        xj/  : :  3  :  5,  j 

J.?is.  X  =  10,  y  =  2. 

14.  Given    x  +  y  =  24    and    xy  '.  x^  •\-  y"^  \\  ?i  -.  10,  to    find  x 
and  ?/.  -47ZS.  X  =  18,  ?/  =  6. 

15.  Given    x  :  y  : :  3  :  2  and  x  +  6:y~6::3:l,  to  find   x 
and  y  Ans.  x  =  24,  y  =  16. 

16.  Given    xy  —  320  and  x^  —  y  :  (x  —  y)^  : :  61  :  1,  to  find 
X  and  y.  ^»s.  x  =  20,  y  =  16. 

ARITHMETICAL   PROGRESSION. 
225.  A   Series  is  a  succession  of  terms,  each  of  which  is  de- 
rived from  one  or  more  of  the  preceding  terms,  by  a  Jixed  law. 

1,  3,  5,  7,  9,  &c. 

is  a  series  in  which  any  term  is  derived  from  the  preceding  one 

by  adding  2. 

3,  6,  12,  24,  48,  &c. 

is  a  series  in  which  any  term  is  derived  from  the  one   preceding 

by  multiplying  by  2. 

1,  3,  4,  7,  11,  18,  &c. 

is  a  series  in  which  any  term  is  found  by  adding  the  two  preced- 

'ng  it,  after  the  second  term. 


230  ARITHMETICAL    PEOGRESSION. 

22€i.  An  Aritlimetical  Progression  is  a  series  whose  law  is 
that  any  term  is  found  h)/  adding  a  constant  qiiantify  to  the  'pre- 
ceding term. 

The  common  difference  is  the  constant  quantity  to  be  added. 

The  progression  is  increasing  when  the  common  difference  is 
positive. 

The  progression  is  decreasing  wlien  the  common  difference  is 
7iegative. 

The  number  of  terms  of  a  series  may  be  limited  or  infiuite. 

The  first  term  is  that  with  which  the  progression  commences. 

The  last  term  is  that  with  which  the  progression  is  supposed 
to  terminate. 

The   sum  of  tJte  terms  is  the  amount  of  all   the   terms  of  the 

progression. 

3,  7,  11,  15,  19, 

is  an  increasing  arithmetical  progression,  in  which  3  is  the  frst 
term^  4  is  the  common  difference,  19  is  the  last  term,  5  is  the 
numher  of  terms,  and  55  is  the  simi  of  the  terms. 

19,  15,  11,  7,  3, 

is  a  decreasing  arithmetical  progression,  in  which  19  is  the  first 
term,  —  4  is  the  common  difference,  3  is  the  last  term,  5  is  the 
number  of  terms,  and  55  is  the  siim  of  the  terins. 

4,  3,  2,  1,  0,  ~  1,  -  2,  -  3,  -  4, 

is  a  decreasing  arithmetical  progression,  in  which  4  is  the  first 
term,  —  1  is  the  common  difference^  —  4  is  the  last  term,  9  is  the 
number  of  terms,  and  the  sum  of  the  terms  is  0, 

22*T,  To  find  the  last  term,  when  the  first  term,  number  of 
terms,  and  the  common  difference,  are  known. 

Let  I  =  last  term,  a  =  first  term,  n  =  number  of  terms,  and 
d  :^  common  difference',   then  the  progression  will  be 

a,     a  -f  d,     a  +  2<^7,     a  +  Zd,     a  +  4c7,     a  -j-  hd,  &c. 


ARITHMETICAL     PROGRESSION. 


231 


In  wliich  any  of  the  numerical  coefficients  is  represented  by  ti  —  1. 
Therefore,  I  =  a  -{■  (?i  —  1)  cl  (1) 

If  d  is  negative,  then     I  =  a  —  (u  —  1)  d.  (2) 

2.  To  find  the  sum  of  the  terms,  when  the  first  term,  the  last 
term,  and  the  number  of  terms,  are  known. 

Let  s  =  sum  of  the  terms,  I  =  last  term,  a  =  JiJ-st  term,  and 
71  =  number  of  terms;  then,  writing  the  progression  in  a  direct 
and  reverse  order,  we  have  the  equations — 


s  =  a  -\-  a  -{-  d  -{-  a  -{-  2d  -\-  a  -}-  Sd  -{■ /. 

s  =  ^  +  I  —  d  +  I  —  2d  +  I  —M  + a. 


By  addition,  2s  =  a-{-l-{-a-]-l-\-a-{-l-{-a-{-  I  -{• 


a  -\-  I,  in  which  a  -{-  I  is  taken  as  many  times  as  is  indicated  by 
n,  the  number  of  terms. 


Therefore,         2s  =  (a  +  Z)  ??,  or  s  = 


(a  +  I)  n 


(3) 


From  equations  (1)  and  (3)  the  following  table  is  easily  made : — 


No. 
1. 
2. 

3. 
4. 

5. 
6. 

7. 
8. 

9. 
10. 

Given. 

Unknown. 

Values  of  the  Unknown  Quantities. 

a,  d,  n, 

I,  s, 

I  =1  a-\.  {ji  —  l)  d;             s  =  \n\2a-\-{n  —  l)d\ 

a,  d,  I, 

71,  S, 

n  =  \{l-a)^l;             .  =  1  (Z+«)  [l-a+d). 

a,  d,  s, 

71,  I, 

71  = ^— —  '^       ;     Z  =  a  +  (n  —  1)  d. 

a,  n,  I, 

s,  d. 

<f    ^  11  ( n    \  ■  1\  •                          d  

a,  n,  s, 

d,l, 

J        2  (5  —  an)                       ^          2s 

71  {?l- —  1  )  '                                                71 

a,  I,  s, 

n,  d, 

,,  _     2'^    .                                          {l+a){l-a) 

~  a-\-r                                   —2s—{l-\-  a)  ' 

d,  71,  I, 

o,s, 

a  =  l  —  {7i  —  l)  d;              s  =  ^7il2l—(n  —  l)d]. 

d,  71,  s, 

a,  I, 

2s  —  7i{n—l)d                    2s-^7i{n  —  l)d 
""^      '          2n               '         ^=                2n 

d,  I,  s, 

71,  a, 

2l-\-d±\/(2l-^d)^  —  8ds             .      .         ,,, 

n  =  ■ i ! — '- cr  =  Z — in  —  Y)d. 

2d                                         ^           ' 

n,  I,  s. 

a,  d, 

n           '                                      71  [n  —  1) 

232  ARITHMETICAL     PROGRESSION. 

EXAMPLES. 

228.  1,  If  a  =  1,  d  =  2,  what  is  the  sum  of  n  terms  of  the 
progression  ? 

Here   s  =  J  ?z  [2a  +  (?t  —  1)  d'],  i.e.  J  n  [2  +  (n  —  1)2],  or  s  =  ?i2. 

2.  What  is  the    sum  of  25    terms  of  the    progression  1,  8,  5, 
7,  9,  &c.  ?  Ans,  625. 

3.  If  a  =  1  and  d  =  1,  what  is  the    sum  of   n    terms  of  the 

progression  ?  .  .        71  (n  -\-  1) 

Ans.  — ^^ — -' 

4.  How  many  strokes    does    the   bell  of  a    clock    make   in    12 
hours  ?  Ans,   78. 

5.  If  a  =  2,  I  =  29,  and  d  =  3,  find  n  and  s. 

Ans.  n  =  10,  s  =  155. 

6.  If  a  =  5,  d  =  6,  and  s  =  2945,  find  /^  and  n. 

Ans.  n  =  31,  ?  =  185. 

7.  If  a  =  5,  Ji  =  31,  and  I  =  185,  find  d  and  s. 

Ans.  cZ  =  6,  s  =  2945. 

8.  If  a  =  1,  n  =  14,  and  s  =  196,  find  d  and  I. 

A71S.  ^  =  2,  ?  =  27. 

9.  If  a  =  1,  Z  =  20,  and  s  =  210,  find  n  and  c?. 

J.WS.  d  =l,n  =  20. 

10.  If  cZ  =  3,  n  =  21,  and  I  =  70,  find  a  and  s. 

.Itis.  a  =  10,  s  =  840. 

11.  If  c?  =  3,  7i  =  21,  and  s  =  840,  find  a  and  Z. 

J.71S.    a  =  10,  I  =:  70. 

12.  If  <Z  =  —  4,  ?  =  12,  and  s  =  72,  find  a  and  n. 

Ans.  a  =  24,  Qi  =  4  or  9. 
The  series  is  24,  20, 16, 12,  or  24,  20, 16, 12,  8,  4,  0,  —4,  —  8. 

13.  If  n  =  21,1=  70,  and  s  =  840,  find  d  and  a. 

Ans.  d  =  S,  a  =  10. 

14.  Insert  four  arithmetical  means  between  1  and  16. 


ARITHMETICAL     P  H  0  G  R  E  S  S  I  0  N.  233 

Since  the  means  are  4,  the  number  of  terms  must  "be  6. 

Hence  the  formula    d  =  =-    becomes  d  =  — ^ r  =  3. 

?i  —  1  b  —  1 

.-.    The  series  is     1,  4.,  7,  10,  13,  16. 

5i29.  15.  A  starts  from  a  certain  place,  travelling  1  mile  the 
first  day,  2  the  second,  3  the  third,  and  so  on.  At  the  end  of  4 
days,  B  starts  from  the  same  place,  travelling  uniformly  at  the 
rate  of  9  miles  a  day.     When  will  B  overtake  A? 

Let  X  =  the  time. 

The  distance  travelled  by  A  will  be  — ^^^-— (^Vide  ex.  3.) 

The  distance  travelled  by  B  will  be  9  (x  —  4). 

Therefore,         "^  ^^  ,"^  ^^  =  9  {x  -  4).    Ans.  x  =8,  or  -9. 

In  8  days  each  will  have  travelled  36  miles.  In  9  days,  45 
miles. 

230.  16.  A  starts  from  a  certain  place,  travelling  1  mile  the 
first  day,  3  the  second,  5  the  third,  and  so  on.  At  the  end  of 
two  days,  B  starts  from  the  same  place,  travelling  uniformly  at 
the  rate  of  9  miles  a  day.     When  will  they  be  together? 

Ans.  In  3  days,  and  again  in  6  days. 

17.  The  sum  of  four  numbers  in  arithmetical  progression  is 
56.     The  sum  of  their  squares  is  864.     What  are  the  numbers? 

Let  the  numbers  be   x  —  y,  x^  x  -\-  y^  x  -\-  2.y 
Then         4x   +  2^  =  56, 
and  4jj2  +  ^xy  +  6/  =  864. 

The  numbers  are    8,  12,  16,  20. 

18.  The  sum  of  four  numbers  in  arithmetical  progression  is 
28.     Their  continued  product  is  585.     What  are  the  numbers? 

Let  the  series  be   x  —  3y,  2;  —  y,  a^  +  y?  ^  +  %• 

Then     \x  =  28,     and     a*  —  lO.ry  +  9/  =  585. 

Ans.  1,  5,  9,  13. 
20 


234  GEOMETRICAL    PROGllESSION. 

331.  A  Geometrical  Progression  is  a  scries  whose  law  is,  that 
any  term  is  found  hi/  multiplying  the  preceding  term  hy  a  constant 
quantity. 

The  Q-atio  is  the  coustant  quantity  used  as  a  multiplier. 

The  progression  is  increasing  when  the  ratio  is  greater  than 
unity. 

The  progression  is  decreasing  when  the  ratio  is  less  than  unify. 

The  Jirst  term,  last  term,  7iumher  of  terms,  and  sum  of  the  tenuis 
are  the  same  as  in  arithmetical  progression. 

1,  3,  9,  27,  81, 

is    a    geometrical    progression,  increasing,  whose    ratio   is   3,  first 
term  1,  last  term  81,  number  of  terms  5,  and  sum  of  terms  121. 

^y   '^>   -^j   2'   4'   8? 
is    a    decreasing    geometrical    progression,  whose  ratio   is    i,  first 
term  4,  last  term  |,  number  of  terms  6,  and  sum  of  terms  7|. 

1.  To  find  the  last  term,  when  the  first  tei-m,  number  of  terms, 
and  ratio  are  known. 

Let  I  =  last  term,  n  =  number  of  terms,  and  r  =  ratio,  a  = 
first  term.     In  this  case  the  progression  will  be — 

a,  ar,  ar'^^  ar^,  ar^,  a?-^,  &c. ; 

in  which  any  exponent  is  represented  by  n  —  1. 

Therefore,  I  =  av^-i.  (1) 

2.  To  find  the  sum  of  the  terms,  when  the  first  term,  number 
of  terms,  and  ratio  are  given.  If,  in  addition  to  the  above  nota- 
tion, s  =  sum  of  the  terms,  then 

s  =  a  -\-  ar  -}-  ar"^  -{-....  +  «^'"~"-^- 

Multiplying  by  r,  rs  =  or  -f  «^'^  +  ar'^~^  +  ar\ 

whence  rs  —  s  =  ar'^  —  a, 

ar""  —  a        a  (r»  —  1) 
and  .  =  -^— ^  =      V-l     •  (2) 


GEOMETRICAL     PR  OGRES  SIGN. 


235 


From  (1)  and  (2)  the  following  tabic  is  readily  formed 


No. 


Given. 


a,  r,  n, 
a,  r,  s, 
a,  ??,  s, 
r,  n,  s, 


a,  r,  n, 


a,  r,  I, 


a,  n,  Z, 


r,  n,  I, 


Required. 


9. 

r,  ??,  I, 

10. 

r,  n,  s, 

11. 

r,  I,  s, 

12. 
13. 

n,  I,  s, 

a,  71,  I, 

14. 

a,  n,  s, 

15. 

a,  ?,  s, 

IG. 
17. 

n,  I,  s, 

a,  r,  I, 

18. 

a,  r,  s, 

19. 

«,  I,  s, 

20. 

r,  I,  s, 

Formulas. 


a  -f  (r 


l{s  — 


1  = 


1) 


—  a  (s  —  a)"'~^ 
(r  —  1)  s?-"-i 

o-J*  —  1 


=  0. 


a 


(7-^  —   1) 

r  — 1 


—  1 


n 


—  a«-i 


Z«— 1  —  a«  — 1 


I  (r^  —  1) 
(r  —  1)  r**-^' 


|.M 1 


(;■  -  1) 


!■"■   —    1 

a  =  r/  —  (j-  —  1)  s. 
a  (s  —  «)'»-i  ^  l(s  —  ly-'^ : 


0. 


Qj^.ti  —  ^.g  _j_  5  —  a  =  0. 

(s  —  0  r'^  —  sr"-i  4-  ?  =  0,     ?•  =      ~   ,• 
^  s  —  6 

(s  —  Z)  r'*  —  sr"-l  +  ?  =  0. 


loo-  ^  __  loe: 


a 


n 


n  = 


n 


Iog:  ?' 


+  1- 


log  [a  +  (r  —  1)  s]  —  iog  a 


log  r 


loo;  I  —  I02:  <^ 


+  1- 


log  (s  —  a)  —  log  (s  —  0 

log  Z  —  log  [rl  —  (r  —  1)  s]    ^    ^^ 

log  >• 


236  GEOMETRICAL    PROGRESSION. 

EXAMPLES. 

1.  If  «  =  5,  T  =  10,  and  n  ==  7,  what  is  the  sum  of  the  series? 

Ans.  55^555, 555. 

2.  If  rt  =  1,  r  =  2,  and  n  =  7,  what  is  the  last  term?    Ans.  64. 

3.  If  a  =  1,  r  =  2,  and  s  =  127,  what  is  the  number  of  terms? 

Ans.  7. 

4.  If  a  =  l,  1=  27,  and  s  =  40,  what  is  the  number  of  terms? 

Ans.  4. 

5    Insert  4  geometrical  means  between  2  and  486. 

1 

Since  r  =  i  —  j         ,  we  have  r  =  1/243  =  3. 

2,  6,  18,  54,  162,  486,  is  the  series. 

6.  Insert  5  means  between  1  and  q\. 

Ihe  series  is,  1,  -^,  ^5  g,  yg,  -g^,  g^. 

-r^  ,     ,«.    .  *"^  —  «         cs  —  rl 

232.  Formula  (6)  is  s  = .-  =  -^ 

^  y  —  i  1  —  r 

If  r  is  a  proper  fraction,  the  progression  is  decreasing;  and  if 

the  series  be  carried  to  infinity,  the  last  term  becomes  0. 

The  formula  will  then  become  s  =  r .         (21) 

1  —  r  ^     ^ 

EXAMPLES. 

1.  Find  the  sum  of  the  series  4  +  i  +  g  +  y g  +  32)  ^^-  *^ 

/=  0. 

i 

Here  a  =  ^,  »'  =  2  :   hence   s  =  :| — ^^  =  1.  Ans.  1. 

2.  Find  the  sum  of  the  series  J  +  ^  +  ^V  +  bV»  <^°-  ^^  infinity. 


1 

3 1 


i-i 


2.  Ans.  ^. 


3.  Find  the  sum  of  the  series  1  4-  4  _{_  ^j^  ^c.  to  infinity. 

s  r=^ r  -^  5.  Ans.  5. 

1  —  i 


GEOMETRICAL     PROGRESSION.  237 

4.  Find  the  sum  of  the  series  |  +  2  +  |j  &c. 

s  —  -,  _  3  —  Y  —  ^3.  ^ns.  ^3. 

5.  What  is  the  sum  of  the  series  1  +  —  +  -^  H — ^'  &c.  ? 

*x!y  Cc/  fcC- 

S  =  r  =  ^.  Alls. 


\  i-  X   —  1*  X  —  1. 

X 

6.   What    is    the   sum  of  the    series    1  +  -.   +  .        ...^  + 


(x4-  1) 

_  1  rc  +  l 


X 


X  +  1 

7.  What  is  the  sum  of  the  series  x  -\-  xf/  -\-  xi/^  +  x^^,  &c. 
whenv/<l?      -  ^,,5.-^. 

PROBLEMS. 

233.  1.  The  sum  of  three  numbers  in  geometrical  progression 
is  14,  and  the  sum  of  their  squares  is  84.  What  are  the  num- 
bers ? 

Solution. 

Let  the  numbers  be  a:,  x?/,  and  a^/^ 

Then  x -\-  x?/  -}-  xif  =  14,  (1) 

and  x"  +  xhf  +  xh/  =  84.  (2) 

Divide  (2)  by  (1),  and  we  have 

X  —  ^y  +  ^y"^  =  6-  (^) 

Subtract  (3)  from  (1),  and  we  have 

2.r?/  =  8,  or  xy  =  4.  (4) 

4 
For  X  place  —  in  equation  (1),  and  we  have 

i  +  4  +  4^  =  14.  (5) 

y 

whence  7/  =  2,  or  \. 

X  =  2,  or  8. 
And  the  series  is     2,  4,  8,  or  8,  4,  2. 


238  GEOMETRICAL    PROGRESSION. 

2.  The  sum  of  three  numbers  in  geometrical  progression  is  21, 
and  the  sum  of  their  reciprocals  is  -j^^-     What  are  the  numbers? 


X  -\-  Xf/  -j-  xy"^  =  21. 

(1) 

-  +  -+  ^ -  - 

X        ory        xy^         ^  ^ 

(2; 

whence 

1  +  ^   -\-y'   =~r. 

(3) 

and 

1+y  ^f  ='< 

(4) 

.'. 

21        -ixf      .                     ^ 
—  =       ';:  ,  whence  xy  =  b. 

X                1^ 

Substitute  the  value  of  xy  in  (4),  and  we  have 

i  +  ,  +  ,^  =  §^. 

whence  y  =  2   or  1. 

a;  =  3   or  12. 
And  the  series  is  3,  6,  12,  or  12,  6,  3. 

3.  The  product  of  three  numbers  in  geometrical  progression  is 
64,  and  the  sum  of  their  cubes  is  584.     What  are  the  numbers? 

The  equations  are     x  X  xy  X  xy^  =  64, 
and  x^  +  x^y'"^  -f  x^y^  =  584. 

The  numbers  are   2,  4,  8. 

4.  The  sum  of  the  first  and  last  of  three  numbers  in  geo- 
metrical progression  is  52.  The  square  of  the  mean  is  100. 
What  are  the  numbers?  Ans.  2, -10,  50 

5.  The  sum  of  the  first  and  second  of  four  numbers  in  geo- 
metrical progression  is  15.  The  sum  of  the  third  and  fourth  is 
60.     Wliat  are  the  numbers? 

X  -\-  xy  =  15,     and     xy^  -f  ^2/^  =  60. 

Ans.  5,  10,  20,  40. 

6.  The  sum  of  the  extremes  of  four    numbers    in    geometrical 


INDETERMINATE    COEEFICIENTS.  239 

progression  is  84.     The  sum  of  the  means  is  36.     What  are  tho 

numbers  ? 

xy  -\-  xy^   =  36,     and     x  +  xy^  =  84, 

or,        xy  (1  -f  y)  =  36,     and     x  (\  -\-  y^)  =  84. 

whence        ~~  (1  —  y  +  y)  =  -|-        y  =  3,  or  J. 

The  series  is  3,  9,  27,  81. 

7.  The  difference  between  the  second  and  fourth  of  four  num- 
bers in  geometrical  progression  is  24.  The  sum  of  the  extremes 
is  to  the  sum  of  the  means  as  7  to  3.     What  are  the  numbers  ? 

Ans.  1,  3,  9,  27. 

8.  What  is  the  third  term  of  a  harmonical  proportion  whoso 
first  -and  second  terms  are  12  and  15  ? 

If  X  represent  the  required  term,  we  have 

12  :  cc  : :  15  —  12  :  X  —  15.     (^Vide  Def.  8.) 
whence  x  =  20. 

INDETERMINATE   COEFFICIENTS. 
234.  Every   identical   equation  containing    but    one   unknown 
quantity  can  be  reduced  to  the  form  of — 

p  -\-  qx  -\-  rx^  -\-,  &c.  =  p^  -\-  q^x  -f  r^x"^  +;  <^c.     (1) 
Or,  by  transposition,  to  the  form — 

(P  -  P')  +  (?  -  20  ^  +  (>'  -  r^)  ^'  +;  &c.  =0.    (2) 

S35.  From  the  nature  of  an  identical  equation  (vide  92,  5), 
equations  (1)  and  (2)  must  be  true  for  all  possible  values  of  x; 
that  is,  we  may  take  a;  =  ^  =  0,  1,  2,  3,  &c.  ]  and  what  is  true 
of  the  coefficients  when  x  equals  any  one  of  these  values,  is  true 
when  any  other  value  is  taken  for  x. 

236.  Because  the  coefficients  of  the  different  powers  of  the 
unknown  quantity  in  equation  (2)  are  coefficients  of  indeter- 
minate quantities,  they  are  called  Indeterminate  Coefficients, 


240  INDETERMINATE    COEFTICIENTS.  '    , 

I 
23'^.    In  any  identical  equation,  containing  but  one    unknown       \ 

quantity,  the  coefficients  of  tlie  like  powers  of  this  quantity  in  the       i 
two  members  are  equal   to    each    other.     For,  if  a:  =  0    in    equa- 
tion (1),  the  equation  reduces  to  ] 

P  =  i^'-  I 

If  now  these  quantities  are  cancelled,  we  have  j 

qx  +  1'^^  +;  &c.  =  q^x  -f-  T^x"^  -j-;  &c. 
Or,  by  dividing  by  x,  i 

q  +  rx  -[-,  &c.  =  q^  -{-  r^x  -{-,  &c. 
Now  make  x  =  0,  and  we  have 

i 
In  the  same  way  we  may  obtain  i 

■■  1 
238.  To  develope  an  expression  by  means  of  the  principle  of 
indeterminate  coefficients.  j 

1.  Assume  the  expression  to  be  equal  to  a  series  of  the    form 

p  -\-  qx  -\-  rx^  +  sx^  -f ,  &c.  i 

2.  Clear    the    equation  of  fractions,  or  raise  it  to  the  required 
power. 

3.  Equate  the  coefficients  of   the    like  powers  of   the    unknown       j 
quantity. 

4.  Find  from  these  equations  the  values  of  p>,  q,  r,  &c. 

5.  Substitute  these  values  in  the  assumed  development.  i 

I 

EXAMPLES.  I 

,    ^      ,  1  +  2x      .  .  I 

1.  Develope  :; :  into  a  series.  1 

1  —  X  —  x^ 

Operation. 
1  +  2x 


=  p>  -\-  qx  -f  ^'-^'^  -^  sx'^  -f  ^1^,  &c. 


INDETERMINATE    COEFFICIENTS. 


24:1 


Clear  this  of  fractious^  and  we  have 


1  -{-  2x  =  p  -\-  q   X  -{-  7 


—  P 


—  9. 

—  P 


X-  +  s    x""  -\-  t 


—  r 

—  9. 


—  s 

—  r 


X*,   &c. 


Equating  the  like  powers  of  x,  we  have 

(\  —  p^  3.0  _  Q^  whence  p  =  1. 

0  =  r  —  q  —  Pj  "        r  =  4. 

0  =  s  —  r  —  2-,  ^^        s  =  7. 

0  =  i  _  s  —  7-,  "        ^  =  11 

1  +  2x 


Hence 


2.  Develope 


1  —  X  —  x^ 

1  —  X 


==  1  4-  3x  +  4^2  -f  7x'  +  llo;*,  &c. 

(  Vide  70,  ex.  26.) 


1  +  ^  +  ^ 


-  into  a  series.     (FtVZe  TO,  ex.  25.) 


3.  Develope  — "-^ -r into  a  series.     {Vide  70,  ex.  23.) 

2  4-  3x 

4.  Develope  7^ — ~ -. — ^-1;  into  a  series. 


8  4-  4:^;  +  bx' 


X        34^2        121a;^     „ 
Ans.  1  +  -  _  -^  +  -^.  &c. 


5.  Develope  y  1  —  x^   into  a  series. 

Operation. 
|/l  —  x^  =  p  -\-  qx  -\-  rx^  +  sa;'  +  ^-^S  &C- 
Square  both  sides,  and  we  have 


1  —  x^  =  p^  +  pq 
pq 


X  +  pr 

x^  +_ps 

a;2  -{-  pt 

+  2^ 

-f  qr 

+  ^s 

+  pr 

+  2^' 

_j_  ,.2 

+  i^s 

+  2S 

.^*+,  &c. 


21 


242  INDETERMINATE     COEFFICIENTS. 

Equating  the  proper  coefficients^  Ave  have 

jp2  _.  1^  whence  p  =  1. 

2pq  =  0, 
2pr  -\-  q-  =i  —  1, 
2ps  +  2qr  =  0, 
2pt  +  2qs  +  7-2  =  0, 

2  ^.■t  ^,.6 


cc 

5  =  0. 

u 

r  =  — 

1 

u 

s   =  0. 

u 

t    =  — 

1 

8 

X'         a. 


Therefore,  V\  —  x^  =  1  —  tt  —  V  —  tt^  —  ?  ^^• 

^  2         8         lo 


{Vide  165,  ex.  9.) 


6.  Develope  yl  —  x  into  a  series. 


^4ns.  1  _  -  _  -^-^  _  ^-^-^  -  0-747678  -'  •^^- 

7.  Develope  y  1  -f-  x  into  a  series. 

.7:  x^  3x^  S .  5x* 

Ans.  1  +   2  -  "274  +  27176  ~  2.4.6.8' 

8.  If   X  =^  \    in    the    last   example,  to  what    does    the    answer 
reduce?  Ans.  i/2  =  1.41421. 


339.  1.  Develope  the  expression  (a  -f  ^)'*  ii^^o  a  series. 

We  have  (a  +  Z>)»"  =  a*"  (1  +  — )"'. 

For  convenience  make  x  =  — ,  and  the  expression  becomes 

a  *- 

cC^  (1  +  xY, 

If  now  we  develope  (1  -f  xy^,  and  then  restore  the  value  of  x 
and  multiply  by  a"*,  we  shall  have  the  development  of  (a  +  ?>)"•. 

Let     (1  +  xY  =  p  4-  ^'^  4-  ^'-f'  +  sx^  +  ^-^S  &c.     (1) 
If  now  X  =  0,  this  equation  becomes 

l""-  =  p  ;  that  is,  p  =  1 ; 
whence     (1  +  x)""  =  1  +  IZ-c  +  ^'-^^  +  s.r'  +  ^.r^      (2) 


INDETERMINATE     COEFFICIENTS. 


243 


Since  (2)  is  identical^  we  may  have  x  ==  y  ; 

whence     (1  +  ^)-  =  1  +  ^^  +  rf  +  sif  +  tyK       (3) 
Subtract  (3)  from  (2),  and  divide  by  x  —  y,  and  we  have 

(14-  z)"»  —  (1  -f  vY  _q  [x  —  ?/)       r  (3;2  —  .v^)      s  (x^— ?/3)       ^  (a:*— .V*) 
(l  +  zj  — (1  4-?/)    ~     x  —  y  x  —  y  x  —  y  x—y 

Let  now  x  =  y,  whence  \  -]-  x  =  \  -\-  y,  and  we  have 

m  (1  +  cr)"^-!  =  ^  4-  2ra:  +  Ssa;^  4-  4;x',  &c.         (5) 
Mukiply  both  sides  of  (5)  by  1  4-  -^7  and  we  have 


(4) 


m  (1  4-  a:)"*  =  ^  4-  2r 


:c2  4-  4i  I  x\  &c.      (6) 

+  8s 


x  -\-  Zs 
+  2r 

Multiply  both  sides  of  (2)  by  m,  and  we  have 
m  (1  4-  ^)'"  =  m  -j-  ??i2x  -{-  mrx'^  -\-  msx^  -\-  mtx^j  &c.  (7) 


x-j-3s 


Zr 


a;2+4^ 
3s 


a;'  . . .      (8) 


Hence   ??i+m2a:-|-W2rx^4-w^^^°4--  ■  •■==(l-{-2>r 

Equate  the  coefficients  of  the  like  powers  of  x,  and  we  have 
m  =  q,  whence    q  =  m. 

_  m  (in  -  1) 


mq  =  2r  +  q, 
mr  =  3s  4"  2r, 
qns  =  -it  -\-  OS, 


1.2 

m  (ill  —  1)  (m  —  2) 
1.2.3  ' 

7?i  (m  —  1)  (m  —  2)  (m  —  3) 


1.2.3.4 

Substitute  these  values  in  (2),  and  we  have 

m(m—l)  ,,  mhn—l)(m—2)  ,  ,  m(m—l)(m—1)(m—^    . 

(l4-x)-=l+?«x4--A__Jx2+_L__^A_ _^,;34.^ r2T3T4 ' 

&c.         (9) 

Substitute  for  x  its  equal  — ,  and  reduce,  and  we  have 

a 

((/4i/"  =  rt*"+?/ia'"-iZ-^ \  >'am-2^,2_^  _!^.^^ — A_ y  a^-sis^  &c.    (10) 

after  multiplying  both  sides  by  a"*. 

This  last  equation  is  the  Binomial  Theorem,  and  it  is  true  for 
any  value  of  m  whatever. 


244  INDETERMINATE     COEFEICIENTS. 

I 

2.  Develoi3e  (1  -{-  a-)^  into  a  series.     Here  m  =  ^. 

Substitute  1  for  a,  and  x  for  Z»,  in  equation  (10). 
^«..  (1  +  :.).  =  1  +  -  _  ^-^  4-  ^^-^  _  ^-^-^,  &c. 

3.  Develope  (1  -\-  x)—"^  = into  a  series.     Hero  m  =  —  1. 

^?2S.   (1  +  a:;)~i  =  1  —  X  -{-  x"^  —  x^  -}-  x*"  —  x^,  &c. 

4.  Develope  (l-f-x)  — ^  ==    ^^  into  a  series.    Here  011=  — 2. 

(1  +  xy 

Ans.  (1  -]-  x)-^  =  1  —  2x  +  3x2  —  4x3  _j_  5^4^  ^^^ 

5.  Develope  (1 — x^—^  =  y^ ~3  i^^^  ^  series.      Here  m=  — 3. 

^?2s.  (1  —  x)-3  =  1  +  3x  +  6x2  _|_  10x3  +  15x*,  &c. 

6.  Develope  (1  +  x)-  into  a  series.     Here  711  =  |. 

^            ^3                    ox          ox  X  ox  p 

^ws.  (1  +  x)^  =  1  +  —  +  -g To  +  "128"  ""' 

7.  Develope  (a  -f  x)^   into  a  series. 

^1  j/,     .     X  2x2  2.5x3         „     \ 

^ns.(a  +  x)3  =  a3^1+___^  +  ____  ^e.j 

Ifa=l,andx=l,thenTr2=l  +  ]-^+^-^^,&^^ 

Ti.       -,       ^        o  .1        3/:7    T  ,  o     2.4     2.5.8      2.5.8.16     , 
If  a=l,and  x=2,  then  V  — 1  +  3-^  +.^ir9-3X9J2-'  ^^• 

If  a=8,and  x=l,  then  ^9=2  (  l  +  ..,--J^+^-^,  &e.  ) 

8.  Develope  (1  +  ^)~'  into  a  series. 

Ans.  (1  +  x)-^  =  1  —  -  + _-  -f.^  &c. 

THE   TABLE   OF   LOGARITHMS. 

^40.    AVe    propose    to    show  in  the    present  chapter    how  the 
table  of   logarithms    has    been    constructed.     For    this    purpose  it 


TABLE     0¥     L  0  G  A  11  I  T  11  M  S.  245 

will  be  sufficient  if  we  actually  calculate  the  logaritliuis  of  a  few 
of  the  lower  prime  numbers. 

In  the  equation  cv'  =  iV  (1) 

a  is  the  base  of  the  system,  and  x  is  the  logarithm  of  the  num 
ber  A^. 

Assume  a    =  1  -f-  m, 

and  N  =  1  -{-  n-, 

Then  (1  +  mf    =  1  +  «,                          (2) 

and  (1  +  myy  =  (1  +  ^0"                        (3) 
By  the  binomial  theorem, 

(l+r.O-i/==l  +  xym+^y^"-^^-^^..^+^^^"V.^^  j'^^ -"-\  m^-{.,&c.  (4) 

1.2  1 .  2  .  o 

(1  +  »)v  =  1  +  y«  +  -^^ n^  +  "'-"''l':^  "'  +'  S'-       (6) 

Equate    the    right-hand    members    of    (4)    and    (5),    reject    the 
unity,  and  divide  by  ^,  and  we  have 

. ( „ -f  »^ .  „,.  +  (^!^-m^;>-^-) .  ,„,  +, &,. _  „  +  .vzd . „,  + 

iiszm;^„ls.o.)  (0) 

If  now  y  =  0,  we  have 
«;  =  log  A^=  log  (1  +  n)  = fVri-^ — ,    4T  r-     (7) 


Since  m  =  a  —  1,  and  n  =  N —  1,  we  have 
^^    '  ~  (a  -l)-i(«  -1)'+K«  -l)'-i(«  -iy+;&c. 


(8) 


This  equation  contains  the  logarithm  of  N  in  terms  of  N  and 
the  base  3  but,  for  actual  computation,  it  is  necessary  to  modify 
its  form. 

Let  the  reciprocal  of  the  denominator  of  (8)  be  represented  by 
My  and  replace  n  for  N  —  1,  and  we  have 

log  (1  +  7i)  =  31  Qi  —  hi"  +  \ii?  —  {ii^  +,  &c.)  (9) 


21* 


246 


TABLE     OF     LOGARITHMS. 


The  factor  M  is  called  the  modulus  of  the  system.     If  n  were 
negative,  we  should  have 

log  (1  _  n)  =  M  (_  n  —  hi"  —  h^  —  ]n*  — ,  &c.)         (10) 
Subtract  (10)  from  (9),  and  we  have 

1  _i_  1) 
(  Vide  138     .)     log  ^^^  =  2M(^n  +  In'  -f  hi'  +  ]7i'  -\-,  &c.   (11) 

W  e  may  now  assume  n  = -,  whence  z, = 

-^  -i?  + 1  1  —  7i         p 

Then 

^  ^^^  ^  "  ^  ^       \  (2;.+  l)^3(2^+ir  5(2^+1)^7(2^+1)^'         / 

(12) 

In  (12)    make  31  =  1,  and  p  =  1.     Then,  since    log  p  =  log 
1  =  0,  we  have 

The  method  of  summing:  this  is  as  follows : — 


'^  o 


82  =  9 
9 
9 
9 
9 
9 
9 
9 


0.66666686  ~  1  =  0.66666666. 


0.07407407  -~    S  =  0.02469136. 


0.00823045 
0.00091449 
0.00010161 
0.00001129 


0.00000014 


5  ==  0.00164609. 

7  =  0.00013064. 

9  =  0.00001129. 

11  =  0.00000103. 


0.00000125  -^  13  =  0.00000010. 


15 

2 


0.00000001. 


log  2  =  0.69314718. 


In  (12)  make  31  =  1  and  ^;  =  2,  and  we  shall  find 

log  3  =  1.098612. 
2  log  2  =  log  4  ==  1.38629436. 
If^  =  4,  then,  log  5  =  1.60943790. 
Log  5  +  log  2  =  log  10  =  2.30258508. 


TABLE     OF    L  0  G  A  11  1  T  II  M  S.  247 

Logaritlims  calculated  as  above  are    known  as  Napierian  Loga- 
rithms, in  honor  of  Lord    Napier,  their  inventor.     It   is  usual  to 
distinguish  them  by  the  contraction  aYa^).     Thus, 
Nap.  log  10  =  2.30258508. 

We  are  now  to  show  how  common  logarithms  may  be  calcu- 
lated from  them. 

241.  Since  Napierian  logarithms  assume  the  modulus  to  be  1, 
it  follows  that  if  the  Xap.  log.  of  any  numhcr  is  multiplied  hy 
the  modulus  of  any  other  system,  the  result  will  he  the  log.  of  the 
same  numher  in  that  system. 

242.  To  find  the  modulus  of  the  common  system  cf  loga- 
rithms. 

Since  log  (1  -f  n)  =  il/  |  «  —  ^  -f  ^  —  j,  &c.  j 

n^        u^        ?{■* 
and      Nap.  log  (1  4-  ''■)  =  '^  —  t,"  "^  ~ 1'  ^^^' 

iU  O  "x 

.                     ir             ^'"^  (^  +  ^0 
we  have  Jl  =  vj -. —. — -, -• 

^ap.  log  (1  -T-  n) 

That  is,  the  modulus  of  any  system  is  the  log.  of  any  number 
in  that  system,  divided  by  the  Nap.  log.  of  the  same  numher. 

The  log.  of  the  base  of  any  system  is  L  Therefore,  the  base 
of  the  common  system  being  10,  we  have 

.r  log   10  L_ 

~  Nap.  log  lU         2.30258508  ' 
Hence,  the  modulus  of  the  common  system  is,  0.4342944:819. 

243.  If  now  the  Nap.  logs,  of  2,  3,  4,  5,  &e.  are  multiplied 
by  0.4342944819,  we  shall  have 

Common  log.  2  =  0.301030. 
"  '^     3  =  0.477121. 

"  "     4  =  0.602060  =  2  log  2. 

"  "     5  =  0.698970. 

c^vC.  &C. 


248  TABLE     OF     LOGARITHMS. 

In  this  -vvaj  tte  whole  table  may  be  constructed.  In  practice, 
the  modulus  may  be  retained  in  the  formula  to  save  the  trouble 
of  multiplication;  and  the  decimals  may  be  carried  to  any  de- 
sirable extent.     (  Vide  153.) 

244.  To  find  the  Napierian  base. 

From  242  we  must  have  the  following  property. 

The  logs,  of  the  same  number  in  different  systems  are  to  each 
other  as  the  moduli  of  the  respective  systems. 

If  a  represent  the  base  of  the  Napierian  system,  then,  since 
its  log.  is  Ij  and  the  modulus  of  the  system  1,  we  have 

com.  log  a  :  1  : :  0.4342944819  :  1. 

Multiply  extremes  and  means,  and  we  have 

com.  log  a  =  0.4342944819. 

The  number  in  the  tables  corresponding  to  this  log.  is 

2.718281828459, 

which  is  the  base  of  the  Napierian  system. 

S45.  To  construct  a  table  of  logarithms  according  to  any 
system  whatever. 

In  the  equation  cv'  =  JSf,  assume  a  equal  to  the  desired  base, 
and  N  equal  to  the  consecutive  numbers,  and  resolve  the  result- 
ing equations.     Thus, 

If  we  desire  the  base  to  be  2,  make 

2'  =  1,  2==  =  2,  2'  =  3,  2=^  =  4,  2*  =  5,  &c. 
and  resolve  the  equations.     Thus, 

1.  X  log  2  =  log  1,  whence  x  =  J-g  =  gjj^  =  0.000000. 

2.  .log  2  =  log  2,       "       .  =  g  =  |lg=  1.000000. 


PRACTICAL    APPLICATIONS.  249 

log  3         477121         .  r,Qif;^Q 
3.     rr  log  2  =  log  3,  whence  ^  =  j^  =  301080  =  ^'^^^^^^ 

o 
&C.  &C. 

By  continuing    tlie    process,   we    should    form    a   table  of  logs, 
with  2  as  a  base. 

It  is  evident  that  the  base  cannot  be  a  negative  numler. 

PRACTICAL   APPLICATIONS. 

246.  1.  Solve  the  cr|uation  7^  =  13. 

log  13  01Q1 

log  ( 


log  2  -  log  3 
log:  o  —  loo;  0 


2.  Solve  the  equation  ^ 


8.  Solve  the  equation  onn''  =  a. 

I02:  ct  —  log  m 

Ans.  X  =--  — ^= — -, 

log  n 

4.   Find  the  value  of   n  in  the  equation  I  =  «r"-i. 
log  I  =  log  a  +  («  —  1)  log  r. 


whence         n  —  1  = 


log  I  —  iVg  '«- 
I02:  r 


Therefore,  n  =  '""  '~  '°^  "  +  1.  (  Vide  231,  form.  17.) 

y  log  r 

CT  (?•"  —  1) 

5.  Find  the  value  of  n  in  the  equation  s  =  — ^TZTl — 

We  have  s  (r  —  1)  =  ar""  —  a, 

or,  a  -\-  (r  —  1)  s  =  ar'", 

whence  log  [a  +  (r  —  1)  .s]  =  log  ^y  +  ^^  log  r. 

log  la  +  (/•  —  1)  -s]  —  log  « 
Therefore,       n  = j^^^;  ' 

(Fi(7e  231,  form.  18.) 


250  PRACTICAL    APPLICATIONS. 

6.  Find     tlie    value    of    n    in    the     equation    a  (s  —  a)^~^  = 

\Ye    have       log  a  -f  0'  —  1)  log  (s — l)  =  log  l-^Cn—l)  log  {s  —  I). 

mi        ^  I02:  ^  —  log  a 

Therefore,      n  =  . ^^^ .-^— -f  1. 

log  C«  —  a)  —  log  is  -  I) 

Vide  231,  form.  19.) 

7.  Find  ?i.  in  the  equation  (s  —  ^)  ?■"  —  fe-r"— ^  -f  ?  =  0. 
This  is  the  same  as  r  (s  —  l}?-''-^  —  s/'"-i  -f  ?  ==  0, 
which  is  [r/  —  (r  —  1)  .s]  7''  — 1  =  /. 

The  log  of  this  is    log  [rl—  (r  —  1)  .s]  +  (n  +  1)  log  r  =  log^. 

losr  I—  log  [rZ  —  (?'  —  1)  .si 

whence         n  =  -^ ^ !^ ^— ^  +  1 

log  r 

Vide  231,  form.  20.) 


CHAPTER     VIII. 

EQUATIONS  OF  THE  THIKD  DEGREE. 

a47.  The  general  form  of  equations  of  tliis  degree  i^ 

cc'  -}-  px^  -{-  2'X  =  m.  (1) 

In  which  p,  q,  and  m  may  be  positive  or  negative. 

24S.  If  p  =  0,  (7  =  0,  and  m  =  «%  we  have 

a:^  =  a^j  or  a.^  —  a^  =  0. 
By  78,  this  equation  may  be  written  thus: 
(.r  —  «)  (.i'^  +  ax  +  «")  =  0. 
Divide  by  each  factor  of  the  first  member,  and  we  have 

X  —  a  =  0  and  x-  +  ax  -f  a-  =  0. 
From  the  first  of  these  we  have 
X  =  a;  and  from  the  second,  a:  =  ^  (  --  1  ±  |/_  y).     (  FiWe  167.) 

Li 

Hence   the  equation  has  three   roots,  two  of  which  are  imagi- 
nary. 

249.  If  p  =  0,  (^  =  0,  and  m  =  —  a^  wc  have 

a.-^  =  —  (x\  or  .>:'  -f  <^^^  =  ^• 
By  79.  this  may  be  written  thus : 

{x  -f  a)  0^2  —  ax  -f-  rr)  =  0. 


251 


252     EQUATIONS  OF  THE  THIRD  DEGREE. 


Hence  x  =  —  a,  and  x  =  ^  (1  dz  y"  _  ^S).     (  Vide  167.) 

250.     1.  Given  x^  =  1,  to  find  the  values  of  x. 

Here  x^  —  1  =  0; 

Hence  (x  —  1)  (x^  +  rz:  +  1)  =  0; 

Whence  x=l  and  a:  =H.  —  1  ±  V  —  3). 

2.  Given  x^  =  8,  to  find  the  values  of  x. 


Ans.  X  =:  2  and  —  1  it  V —  o. 

3.  Given    a;^  ==  27,    :r«  =  64,    x^  =  —125,  and  x^  =   —216, 
to  find,  &c. 

4.  Given  x^  =  10,  to  find  the  values  of  x. 

Alls.  X  =  i^Tu  and  #"  y^  (  —  1  ±  l/"^^^), 
or  X  =  2,1544  and  —  1.0772  it  1.8657  V^^. 

251.   If,  in  the  equation  x^  -\-  px^  -\-  qx  -{■  m  =  0,  a  is  an 

EXACT    ROOT,    THEN    THE   FIRST    MEMBER   IS   EXACTLY   DIVISIBLE 
BY  X  —  a. 

For,  since  a  is   an   exact   root,  wc   may  substitute   it   for  x  in 
the  given  equation,  and  write 

a'  +  i^"^  +  S'^  +  *'^  =  0.  (2) 

Now  let  us  proceed  to  divide  the  given  equation  by  x  —  a. 

0?  -j-  pj^  -\-  qx  -\-  in\x   —  a 

x^  —  ax^  x^  -|-  (p  +  «)^  +  2'  +  <^  (p  +  <^) 

(p  4"  ci^x"^  —  a  (p  -j-  a)x 

{q  -{•  a  {p  +  «)  )'^  +  ^^ 
(<7  +  a  (^3  +  a)  )x  —  o'5'  —  a^  {p  -\-  a) 
Tlie  last  remainder  is  aq  -\-  a?  (^p  -\-  a)  -\-  m  =■  d?  -\-  pa}  -\-  qa  -{-  m. 

But,  by  (2),  this  remainder   is  0.      Therefore   the   quotient  is 
exact. 


EQUATIONS    OF    TUE    THIRD    DEGREE.  253 

252.  It  is  evident  that  the  same  reasoning  applies  to  any 
equation  of  the  form 

^n  _j_  ^^^n  _  1  _^  ^^^n  -  2  &c.    +  fX  +  111  =  0.  (3) 

For,  since  the  dividend  is  equal  to  the  divisor  multiplied  by 
the  quotient  plus  the  remainder,  if  we  denote  the  quotient  by  /, 
and  a  supposed  remainder,  on  division,  by  r,  we  should  have 

X^  -f  px''  - 1  -f-  qx""  -  2  &C.   +  tX  -\-  7)1  =  (^x  —  a)  I  -\-  v.  (4) 

Now,  if  a  is  substituted  for  x,  the  first  member  of  this  equa- 
tion is  0.  But  the  first  term  of  the  second  member  is  0,  and  .-. 
r  =  0. 

Hence,  if  a  is  a  root  of  any  equation  of  the  form  of 

(3),  THAT   EQUATION   IS   DIVISIBLE   BY   X  —  a. 

EXAMPLES. 

253.  1.  One  root  of  the  equation  x^  —  6^^  +  llx  =  6  is  1 : 

what  are  the  other  roots? 

We  may  write  x^  —  6x^  -f  llx  —  G  =  0,  which,  by  251,  is  ex- 
actly divisible  by  x  —  1.  On  dividing,  the  factors  of  the  equa- 
tion are  found  to  be       {x  —  1)  {pc^  —  bx  -\-  Of)  =  0. 

Hence  x^  —  bx  =  —  6 ; 

From  which  a;  =  3,  or  x  =■  2. 

2.  One  root  of  x^  -\-  4x-  —  7x  =  190  is  5:  what  are  the  other 
roots?  Ans.  J  (  —  9  =h  /^fi). 

3.  One  root  of  x^  —  4a;^  —  llx  =  —  30  is  2  :  what  are  the 
other  roots?  Ans.  x  =■  b  and  x  =  —  3. 

4.  One  root  of  x^  -\-  x^  —  22a;  =  40  is  5 :  what  are  the  other 
roots  ?  Ans.  x  =  —  4  and  x  =  —  2. 

5.  One  root  of  x^  -\-  2x^  =:  IG  is  2  :   what  are  the  other  roots? 

Ans.  2  (_i±i/^in:). 

G.   One  root  of  x^  —  2x^  —  2x  =  —  4  is  2:  what  are  the  other 

roots?  Ans.  ±  1.41421. 

22 


25-i      EQUATIONS  OF  THE  THIRD  DEGREE. 

7.  One  root, of  jc^  —  2x  =  —  3  is  I :  what  arc  the  other  roots? 

Ans.  1.0962  and  —  1.5962. 

8.  One  root  of  x^  —  6x-  +  x  =  —  28  is  4 :  what  are  the  other 
roots?  .1;^.'.  3.82841  and  —  1.82841. 

9.  One  root  of  x^  +  ^x-  -f  26x-  =  —  24  is  —  4 :  what  are  the 
other  roots?  Ans.  —2  and  —3. 

254.  If  the  equation  x^  +  jjx^  -\-  qx  -\-  m  =1  0  is  divisi- 
ble  BY   X  —  a,    THEN    a   IS   A   ROOT    OF    THE   EQUATION. 

For,  in  this  case,  the  remainder  is  0,  and,  by  §  251,  we  may 
"Write 

x^  -f  po6^  -\-  qx  4-  'in  =  (cc  —  a)  (x^  -f  (p  ~\-  a^x  +  ^  +  «  (p  +  ^0  )• 

lfx  =  a^  the  second  member  reduces  to  0 ;  .•.  a  substituted 
for  X  will  reduce  the  first  member  to  0.  Hence  a  is  a  root  of 
the  given  equation. 

255.  The  same  troposition  is  true  of  the  equation 
aj"  H- jj.z,"-i  -f  qx''--  &c.  -f  tx  4-  on  =  0. 

For  in  this  case  we  may  write,  by  §  252, 

a"  +  px"  -1  4-  q.r''--  kc.  -f  fx  -f  711  =  (.r  —  a)l 

If  x=-a,  the  second  m.ember  reduces  to  0;  .•.  a  substituted 
for  X  will  reduce  the  first  member  to  0.  Hence  a  is  a  root  of 
the  given  equation. 

256.  Hence,  to  ascertain  if  a  polynomial  containing  x  is  ex- 
actly divisible  by  x  —  a,  substitute  a  for  x,  and  if  the  polijnomial 
reduces  to  0,  the  division  is  exact. 

EXAMPLES. 

1.  The  equation  x^  —  12x2 -{- 47.'c — 60  =  0  has  three  loots 
less  than  10:  what  are  they?  Ans.  3,  4,  and  5. 

2.  The  equation  x^  -f-  ^-^^  +  26x  +  24  =  0  has  three  negative 
rojts  less  than    —10:   what  :;re  thov?      Auft.   —2,  — '1,  and  —4. 


EQUATIONS     OF    THE     T II 1 R  ])     DEGREE.  255 

/ 

3.  The  equation  x^  —  Qx^  +ll:c  —  6  =  0  lias  three  roots  less 
than  10:  what  arc  they?  Ans.  1,  2,  and  3. 

4.  The  equation  x^  +  Ux^  +  44:c  +  32  =  0  has  three  nega- 
tive roots  less  than  — 10:  what  are  they? 

Ans.   —  1,  —  4,  and  —  8. 

25T.     The    equation    x^  -f-p.r-  -\-  qx  -f  ??i  =  0    has    three 

ROOTS,    and    only    THREE. 

For,  if  a  is  one  root  of  the  equation,  its  factors,  by  §  254,  are 

(x  _  a)  Oi-2  +  (j)  +  a)x  -f  5  +  a  (p  +  cv))  =  0. 
Dividing  first  by  one  factor  and  then  by  the  other,  we  have 

X  —  a  =  0,  and  x"^  +  (j^  +  «)-^  +  2'  +  <^  (P  +  ^)  =  ^• 
The  roots  of  which  are  x  =  a,  and 

^  =  -  K(P  +  «)  ±  V  (p  -  ay  -  4  C^r  +  ^j).  (5) 

258.  The  equation  :/;"  A-jpx''-'^  -f  ^x"  — -&c.  -f  ^j?  +  ??i  has 
Qi  roots,  and  only  n. 

For^  if  the  equation  were  of  the  4th  degree,  i.e.  n  =  4,  on 
dividing  by  a:  —  «,  a  being  one  root,  it  would  be  reduced  to  an 
equation  of  the  3d  degree,  which,  by  §257,  has  3  roots;  .-.  the 
equation  itself  has  4  roots. 

Since  the  same  reasoning  applies  to  an  equation  of  the  5th, 
6th,  7th,  &c.  degree,  we    conclude    generally    that  an    equation 

OF    the   7?TH    degree    has    n    ROOTS. 

examples. 

1.  One  root  of  x^  —  llx^  +  1^-^  -f  84  =  0  is  —  2  :  what  are 
the  other  roots? 

Here  a  =  —  2,  j:»  =  -—  11,  and  5  =  16,  and  formula  (5) 'gives 
a:  =  6,  and  a:  =  7. 

2.  One  root  of  x^  +  7^^  —  4x  —  28  =  0  is  —  7  :  what  are  the 
other  roots?  Ans.  x  =  ±  2. 


256     EQUATIONS  OF  THE  THIRD  DEGREE. 

3.  Oue  root  of  x^  +  IZj~  -\-  44.r  +  32  =  0  is  —  1 :  Tvhat  are 
the  other  roots?  Ans.  — 4  and   — 8. 

4.  One  root  of  x^  —  12:c'  +  2Sx'-  -}-  6Sx  —  84  =  0  is  1 :  what 
are  the  other  roots?  Ans.  . 

5.  One  root  of  x'  +  4^^  —  25x2  —  16x  +  84  =  0  is  3  :  what 
are  the  other  roots?  Ans.  . 

6.  One  root  of  x=  +  Ox*  —  bx^  —  141x2  +  4x  +  420  =  0  is  —  5 : 
what  are  the  other  roots?  Ans.  . 

259.  If  a,  h,  and  c  are  the  roots  of  x^  +/>x2  -j-  qx  -\-  m  =  0 , 
then  we  shall  have 

x^  -\-  px^  -^  qx  -\-  m  =  (x  —  a)  (x  —  Z^)  (x  —  c).       (6) 

For,  since  the  given  equation  is  divisible  by  either  of  the 
expressions  (x  —  a),  (x  —  ?y),  or  (x  —  c),  it  must  be  composed  of 
these  three  factors,  and  none  others,  considering,  also,  that  it  can 
have  only  three  roots. 

260.  In  the  same  manner,  if  a,  h,  c,  ....  Ic,  and  I  are  the 
roots  of  the  equation  x"-  -\-  px'"'  —  '^  -]-  qx"-  —  ^  kc.  -\- tx -\- m^  then 
we  shall  have 

x'*  -f  px''  — ^  +  g-x"  —  2  &c.  +  ^x  -f  m  =  (x  —  a)  (x  —  l>)  (x  —  r) 
{x-h)  {x-  I).  (7) 

EXAMPLES. 

1.  Find  the  equation  whose  roots  are  1,  2,  and  3. 

Ans.  (x  —  1)  (x  —  2)  (x  _  3)  =  x3  —  Gx^  +  llx  —  6  =  0. 

2.  Find  the  equation  whose  roots  are  1,  2,  and  —  5. 

Ans.  x'  +  2x2  _  13^  -f-  10  =  0. 

3.  Find  the  equation  whose  roots  are  l/  2,   —  V^,  and  2. 

2x2  _  2x  =  —  4. 


EQUATIONS     OF     THE     T II 1 K  D     DEGIIEE.  257 

4.  Find  the  equation  Tihosc  roots  arc  — G,  — i/8,  and  l/o. 

Alts,  x^'  +  6.t^  —  ox  ==  13. 

5.  Find  the  equation  Avhosc  roots  arc  3,  1  +  yd,  and  1  —  V  o. 

Ans.  x^  —  hx^  -f-  4x  =  —  G. 

261.  If  the  second  member  of  (6)  is  developed  bj  actual  mul- 
tiplication, -^e  have  .i?  -f-  "px-  -f  ^x  -f-  "^  ==  ^-'^  —  (ci  -j-  ^  +  0^^"  "i- 
{ah  -}-  ac  -\-  hc)x  —  ahc.  (S) 

Whence p  =  —  {ci  -\-  h  -{-  c),  q=  ah  -{-  ac  -{-  he,  and  m=  —  ahc. 

(  Vkh  237.) 
That  is, — 

1.  The  coefficient  of  the  first  term  is  1. 

2.  The  coefficient  of  the  second  term  is  the  algebraic  sum  of 
the  roots  with   a  contrary  sign. 

3.  The  coefficient  of  the  third  term  is  the  algebraic  sum  of  the 
roots  taken  in  products,  as  many  times  as  there  are  different  sets 
of  trco  roots  in   each  set. 

4.  The  term  independent  of  x  is  the  product  of  the  roots 
with  a  contrary  sign. 

EXA^IPLE^. 

1.  Find  the  equation  whose  roots  are  1,  2,   and  3. 

Here        j;  =  -  (1  +  2  +  3),  2=  1  X  2  -f  1  x  3  +  2  x  3 

and   m  =  —  1x2x3. 
Ileuce  the   equation   is   x^  —  Gx-  +  H-''  —  G  =  0. 

(T7(7e§  260,  Ex.  1.) 

2.  Find  the  equation  whose  roots  arc  3,  3  +  v'o,  and  3  —  ]    o. 

Here  p  =  — 9,  q  =  24,  and  m  =  — IS. 

Hence  the  equation  is   x^  —  9,/-  +  24x  —  18  =  0. 

3.  Find  the  equation   whose  roots  arc  1,  2,  and   — 3. 

Here  p  =  0^  q  =  —  7,  and  m  =  G. 
Hence  the  equation  is  x'  —  Ix  -\~  G  =  0. 

4.  Find  the  equation  whose  roots  are   —  ^,   — |,  and   — ]. 

22-' 


258  EQUATlO:sS     01-'     TiiE     T II 1 11 D     DEGllEE. 


i;^'    \L   —  ij7    ""^     '"■  —  ^4 


Here  p 

Hence  the  efjiuition  is  x^  A — T~r-{-  — r  -f  Ti  =  ^• 

5.  Find  the  equation  whose  roots  are  5,  5  +  ]/ —  1,   and   5  —      \ 
l/'^TT.  Ans.  x^  —  lox-  H-  76.x-  —  130  =  0.     j 

262.  If  the  second  member  of  equation  (7)  were  developed  by 
actual  multiplication;  we  should  find  the  law  for  the  coefficients  j 
of  the  first  three  terms  the  same  as  before,  while  the  coefficient  j 
of  the  fourth  term  iciU  he  the  algehraic  sum  of  the  j^^'oducfs  of  the  \ 
roots  talcen  os  many  times  as  there  are  different  sets  icith  three  \ 
roots  in  each  set,  the  sign  of  the  final  residt  being  changed.  \ 

The  coefficient  of  the  fifth  term  icill  he  the  algehraic  sum  of  all     ' 
the  p?*oc7wc^s  of  tJie   roots,  taken   as    many  times   as   there   are  sets 
icith  four  roots  in  each  set.  \ 

And,   in    general,  the    cocfiicient    of   the    ni\\    term    will   he   the      ' 
algehraic    sum   of  all   the  products  of  the    roots,   taken    as    many 
times  as  there  are  sets  with  n  —  1  roots  in  each  set,  the  sign  of  the 
final  result  being  changed  when  n  is  even  and  retained  when  n  is 
odd. 

This  n  must   not  be  confounded  with  the  n  which  denotes  the 
degree  of  the  equation. 

EXAMPLES. 

1.  Find  the  equation  whose  roots  are   — 2,   — 2,  4^  and   — 4. 

Here  p  =  4,  (/  =  — 12,  t  =  —  04,  and  m  =  —  64. 
Hence  the  equation  is  x'^  -j-  4:x^  —  VZx"^  —  64  x  —  64  =  0. 

2.  Find  the  equation  whose  roots  are  3,  4,   — 1,  and  — 6. 

A?is.  x*  —  31.t'^  +  42x  +  72  =  0. 

3.  Find  the  equation  whose  roots  are  1,  2,  3,  4,  and  5. 

Ans.  a.-^  —  15.^*  +  Sbx""  —  225x2  +  274x-  —  120  =  0. 

iJ6J$.  The  equation  x^  +7>'"  +  qx  -j-  ??i  =  0  may  be  trans- 
formed   INTO    another   of  the    same   form,  in   v/iiicii   the 


EQUATIONS     OF     THE     THIRD     DEGREE.  259 


ROOTS     ARE     ANY    GIVEN     MULTIPLE     OF     THOSE     OF    THE     GIVEN  i 

EQUATION. 

y  1 

For^  in  the  equation  x"  +  i^t}  +  qx  +  m  =  0,  substitute  j  for  Xy 

1 
and  we  have  73  ^T^  +  T"  +  '^^  =  ^'  which  multiplied  ! 

by  U  gives  y^  +  ^i^^/^  +  J^^9.y  +  ^^^^^  =0;  (9) 

which  is  the  equation  it  was   proposed  to  obtain.      The  roots  of  ' 

(9)  are  h  times  the  roots  of  the  given  equation;  since  y  =  kx. 

I 

EXAMPLES.  I 

1.  Transform   the   equation    aj^  —  6x-  +  11^  —  6  =  0    into    an- 
other whose  roots  are  3  times  as  large. 

?/ 
Here    ic='^,  i.e.  y  =  3x'.     Make  k  of  (9)  =  3,   and   we   have  ! 

y  _  18^2  _^  99^  __  x(52  =  0,  where  p  =  —  G,  ^  =  11,  m  =  —  6.  : 

2.  Transform    the    equation    x^  —  Gx^  +  llx  —  6  =  0  into   an-  , 
other  whose  roots  are  A  as  large.     Here  7c  =  ^. 

Ans.  y  -  3/  +  -^  -  I  =  0. 

I 
264.      In    THE    EQUATION     X^  -\- l^x"^ -\-  qx -\-  77ix'^  —  0,    IF    THE  i 

COEFFICIENTS    ARE    FRACTIONAL,    THE    EQUATION    MAY    BE    TRANS-  | 

FORMED    INTO    ONE    WHOSE    COEFFICIENTS    ARE    ENTIRE.  | 

For,  in   equation  (9)  it   is   manifest   that   k   may  be   so   taken  1 

that   all   the   coefficients   may  be   entire  if  either  j^}  9.)  ^^  *^^  ^^®  ' 

fractions. 

I 

EXAMPLES. 

I 

1x^       Ix         1  I 

1.   Transform   the   equation  a:^  +  -^  4-  ^  +  ^  =  0  into  one  ^ 

whose  coefficients  are  entire.  Here  j;  =  |,  q=i  ■^^,  and  m  =  q\. 
Make  7c  =  8,  and  equation  (9)  becomes  //  +  7^^  +  1^^  -f  8  =  0, 
the  roots  of  which    arc  8  times  those  of  the  given  equation. 


230     EQUATIONS  OF  THE  THIRD  DEGREE. 

2.  Transform   ^^  -h  -j  "f~Tp~f'FT'^^    ^°^^  ^"    equation  wtose 
coefficients  are  entire.     Make  7c  =  4. 

^?IS.  f  -^   if  ^  1/  ^  I  =,  0. 

lice 

3.  Transform    a;'  —  ox^  -J — ^ |  ==  0    into   one  whose    coeffi- 
cients are  entire.      Make  k  =  2. 

Ans.  f  —  Qf  -f  lly  —  6  =  0. 

4.  Transform   x^  —  -rp-  +  ,^^  —  — -  =  0  into   one   whose    coeffi- 

25       30       40 

cients  are  entire.      Make  k  =  150. 

Ans.   f  —  12/  4-  750y  —  84375  =  0. 

265.  The  propositions  of  §§  263  and  264  are  equally  ap- 
plicable   TO    THE    EQUATION    X""  -f  jrx''—'^  -\-  qx""  —  ^  .  .  .  -\-  tx  -}• 

m  =0. 


For,  make  x  =  y,  and  we  have 


^  4-  p^"~    -J-  y^"*"^  ...  +  ^  ^  -f  771  =  0.      Multiply  this  by  ^'*, 

and  we  have  j 

y^  _}_y,y^-l  -f.  qkY~^-  •  •  +  ^^^"''"^  +  ^nk""  =  0.  (9)        ! 

In    this  equation,  7c  may  always   be    so   taken    that   the    fractions 
will  all  disappear. 

EXAMPLES. 


5.r3 


bx-"        7x  13 


1.  Transform     .x* —  -j- .pr-  —  ^r^r^-x   into  one    whose 

b  12        loO       9000 

coefficients  are  entire. 

Make  7c  =  30,  and  vre  have 

/  _  25/  +  375/  —  i:i60y  —1170  =  0. 

The    roots   of  this   equation    are    30    times    those  of   the    given 
equation. 

2.  Transform    :,.-  _4- _-  —  -_- ^  =  0. 


EQUATIONS     OF    THE    THIED     DEGREE.  261 

Make    h  =  GO,  arrd  we  have 
f  ^  65/  +  1891V  —  30720/  —  928800y  +  072000  =  0. 

2S®.      The     equation    x^  +  px^  -\-  qx  -{•  m  =  0    may    be 

TRANSFORMED  INTO  ANOTHER  EQUATION  OF  THE  SA3IE  FORM, 
WHOSE  ROOTS  differ  FROM  THOSE  OF  THE  GIVEN  EQUATION  BY 
A   GIVEN    QUANTITY. 

For,  let  X  =  y  +  ''•  Substitute  this  value  of  x  in  the  given 
equation,  and  we  have 

(y  j^  ry  4-  P  (i^  +  '0^  +  2  (y  +  ^0  +  ^^^  =  0.  By  expanding  the 
terms,  we  have 

,f  _|-  3^v  _j.  Sf/r'-  4-  r^  -\-  j^if  -f  liwy  +  p>^  +  5^  +  'F  +  ^'^  =  ^■ 

Arrange  the  terms  according  to  the  powers  of  y,  and  we  have 
/  +  (3r  +  p)  /  +  (3r2  +  2pr  +  q)u  +  r^  +p?-2  -f  ^r  +  m  =  0  (10). 
Which  is  the  equation  it  was  proposed  to  obtain,  since  y  ^=-  x  —  r. 

If  we  write  the  coefficients  of  this  equation  above  each  other, 
commencing  with  the  last,  we  have 

3r2  +  2pr  +  ^,  (D^). 

If  the  term  independent  of  r  be  considered  as  the  coefficient 
of   r",  these  coefficients  are  derived  in  the  following  way: — 

The  first  coefficient  is  what  the  given  equation  becomes  when 
r  is  substituted  for  x. 

The  second  coffcient  is  derived  from  the  first  by  multiplying 
the  coefficient  of  each  term  by  the  exponent  of  r  in  that  term, 
and  then  diminishing  the  exponent  by  1.  It  is  called  the  First 
Derived  Polynoviiah 

The  third  coefficient  is  derived  from  the  second  in  the  same 
way,  except  that  each  product  is  divided  by  2.  It  is  called  the 
Second  Derived  Polynomial. 


262  EQUATIONS     OF     THE     T  HI  11 1)     DEGREE. 

EXAMPLES. 

1.  Transform  the  equation  x^  —  6x^  -j-  llx  —  G  =  0  into  one 
whose  roots  are  less  than  those  of  the  given  equation  by  4. 

Rere  7-' -}- pr' +  qr -\- m  =    (4)3—    (3(4)2 -f  11(4) —6=    q 
3r2  +  2^r  4-  (2  =  3(4)2  _  i2(4;   +11  =11. 

3r+p==3(4)  -  ^3^  =    6. 

Hence  the  equation  is  j/^  +  Qy^  -]-  1^}/  -\-  Q  =  0. 

The  roots  of  the  given  equation  are  1,  2,  and  3.  Those  of  the 
transformed  equation  are  — 3,   — 2,  and  — 1. 

2.  Transform  the  equation  x^  —  12it;2  -f  47a:  =  60  into  one 
vrhose  roots  are  less  than  those  of  the  given  equation  by  1. 

Ans.  f  —  9/  _f_  26j/  =  24. 

3.  Transform  :i^  —  Ox^  -f  26./:  =  24  into  one  whose  roots  are 
less  than  those  of  the  given  equation  by  1. 

Ans.  y^  —  6^2  _j_  \\y  s=:  G. 

4.  Transform  x^  —  Gj.-  -f  llx  =  6  into  one  equation  whose 
roots  are  less  than  those  in  the  given  equation  by  1. 

Ans.  y^  —  3^^^  -j-  2y  =  0. 

5.  Transform  x:'  -f  13x-  -f  44x  -f  32  =  0  into  an  equation 
whose  roots  are  greater  than  those  in  the  given  equation  by  10. 

{Vide  256,  Ex.  4.)  Ans.  /  —  17/  +  84^  =  108. 

SG^y.  The  proposition  of  §  266  is  equally  applicable  to  the 
equation  x"  +  px'^  —  ^  +  qx^  —  "^  ...-{-  tx  -\-  m  •=  Q.  For,  make 
x  =  1/  ~{-  Vj  substitute  in  the  given  equation,  develope  the  several 
terms  by  the  binomial  theorem,  and  arrange  the  terms  according 
to  the  powers  of  y,  and  the  coefficients  will  become  as  follows: — 

Of  /  it  is     ?•"  +  ^jy«-i  -f  qr^-'^ tr -{-  m  (D^). 

Of  y  it  is  7?r"-i+(?i  — l)pr"-2  +  (n  —  2)qr''-^....  -\-t  (D^). 

Of  /  it  is 

«(»-^^ ,„_2  ,  0—l)C«-2)  „_,      («-2)(«-3) 

1.2  "^  1.2  "'"1.2  ■■\^2)- 


EQUATIONS     or     THE     T II I K  D     DEGREE.  '26o 

Of  /  it  is 

,(,,-1)0,^2)         3       (.  -  1)  (n  -  2)  (.  -  S)  ,^ 

1.2.3  .  1.2.3  ^'^''-      ^-^^^• 

„  -   ^  .    .    ?i(?i  —  1)  (n  —  2)  ( 71—  3)         ,    „  .^  . 

Oh'  It  IS ^  '^"-^  ^  ,  ^^^ --^  v--^  &c.  (DJ. 

Each  of  these  coefficients  is  derived  .from  the  one  immediately 
preceding,  according  to  the  following  taw  of  derived  jiol^iio- 
mials : — 

Multiply  each  term  in  succession  by  the  exponent  of 
r  in  that  term,  divide  the  product  by  the  number  which 
designates  the  place  of  the  coefficient^,  and  diminish 
the  exponent  of  7'  by  unity. 

examples. 

1.  Transform  the  equation  3x*  —  4:X^  +  ^-^^  +  8''  —  12  =  0 
into  one  whose  roots  are  less  than  those  in  the  given  equation 
by  3. 

Here  (D„)  =  3(3^-  4(3y>  +  7(3)-^  +  8(3) -12  =  210, 
(D^)  =  12(3/  -  12(3)-^  +  14(3)  +  8  =    266, 

(D,)  =  18(3)-^- 12(3)   +    7  r=    133, 

(DJ  =  12(3)  -    4  =32, 

(DJ=    3  =        3. 

Hence  the  equation  is,  3/  +  32/  +  133^^  -f  266y  +  210  =  0. 

2.  Transform  x^  —  31.r^  -f  42j;  +  72  =  0  into  one  whose  roots 
are  greater  by  5.  Ans.  /  —  20/  +  119/  — 148y=  216. 

26S.  The  equation  x^  -^  j)x^  -\-  qx  -j-  m  =  0  may  be  trans- 
formed INTO  another  equation  WHOSE  SECOND  OR  THIRD 
TERM    IS    WANTING. 

For,  to  make  the  second  term  disappear,  make,  in  (10), 
3r  +  p  =  0,  or  r  =     ^    . 


261  E  Q  U  A  T  I  0  X  5     OF     THE     T  11 1  L  D     DEGREE. 

To  make  the  third  term  disappear,  make;  in  (10), 


S/-  -f  22^  +  ^  =  0 ;  i.e.  r  =  |(—  p  dr  V  p'  —  oq)- 
EXAMPLES. 

1.  3Iake  the  second  term  disappear  in  the  equation  dS'  —  C..-.^ 
-f  11./:  -6  =  0. 

Here  /•  =  {'  =  2.     Therefore  the  coefficients  of  (10)  become 

r3  ^  pr-  -f  qr  +  m  =     (Tf  —    6(2)2  ^  ii(2)  —  6  =         0. 

Sr^  +  2pr  4-  q  =  3(2)^  -  12(2)  +11  =  -  1. 

3/-+ ^9  =  3(2)   —    6  =0. 

Hence  the  equation  u  ]f  —  y  =^  0. 

The  roots  of  this  equation  are,  of  course,  less  than  those  of  the 
given  equation  by  2.  The  roots  of  the  given  equation  are  1,  2, 
and  3. 

Those  of  the  transformed  equation  ought  to  be  — 1,  0,  and  -fl. 
And  -^e    really  have  y^  —  ,y  =  ,y  (if  —  1)  =  ^;    whence  y  =  0  and 

y  =  =  i. 

It  may  frequently  happen  that  more  than  one  term  disappears 
at  a  time,  as  in  the   example  above. 

2.  31ake  the  third  term  disappear  in  the  equation  x^  —  ^j}  -f- 
11.,-  — 0  =  0. 

Here    r  =:  2  ±  J  i/o.     The  coefficients  of  (10)  become 

(2  ±L  I  Vl'f  -  6(2  ±  \  v'y^y  +  iir2  ±  i  V^)  -  6  =  zp  1 1/3, 

3(2  ziz  I  V%f  —  12(2  ±  \  1/8)  -f  11  =         0, 
8(2  dz  \  VVj)  —    6  =  ±  1/3. 

Hence  the  equation  is  y'  =t  y  Ij.  y-  zp  |  y  3  =  0. 

3.  Make  the  second  term  disappear  in  the  equation  x^  —  14x^ 
-fGlx  =  84.  ,  ,       13y 

o 

4.  Make  the  second  term  disappear  in  the  equation  :>^  —  12.i''' 
-f  47jc  =  GO.  Ans.  y  —  y  =  0. 


2'iX?Ii2i:i£    IJ    it2iiIX23>    JliILT3riJtlJLl.5- 


i^  s:?5SCKr  JT3-  i,.       .  :  - - 

-  7.,.-      2       — — xU  SBB£.    —  _  -1 


;k  4 


%  i^^.  -w^- 


r"  —  3::r"  —  ez  ^  »  =  {r  — 


3* — 3r     *2 —  If*     ,7 —  -»  wi-=zji — - — r  n^ — - — xi   r — ?      /  —  " — aur^ — Si 

—  js          —  -;r-         —  _7«^                  — '  —  —    ■  — XL   r — 2; 

—   _[          —  ff"                  — ~ —  —    -^-i;»fT— 5; 
—  » 


266  L  Q  U  A  L  '  ROOT  3. 

?•'  -f  jyr-  +  qi'  -f  m  =  (r  —  a)  (r  —  Z>)  (r  —  c). 
3,-2  4_  2p>-  +  q=  0'  - «)  0' -  ^)  +  (^-  -  a)  (r  -  c)  +  0'  -  ^)  (^  -  c). 
3r  -f  ^9  =  (r  -  «)  +  (^-  -h)-\-  (r  -  c). 
1  =  1. 
Since  r  is  arbitrary,  we  may  make  it  equal  to  re,  and  thus  obtain 

x^ -\- px'^ -{- qx -\- m  z:=  {x  —  a)  (2;  —  b)  (x  —  c)  (G  ), 

(Di)  =  3z2  +  2i.x-  +  g  =  (.T-a)(.r-i)  +  (:t--«)(z-c)  +  (a:-i)(2--c)(f/,), 

Equation  (d^  proves  that  the  first  derivative  is  equal  to  the 
sum  of  the  products  of  the  several  factors  of  (6)  taken  as  many 
times  as  there  are  different  sets  with  two  factors  in  each  set. 

Equation  (^d^  proves  that  the  second  derivative  is  equal  to 
the  sum  of  the  several  factors  of  (6). 

By  taking  the  same  steps  with  the  equation 

x^-\-px'^^^-\-qx'"—'^. . .  -\-tx-^m=^(^j:—a)(x — 6)(ic — c) (x — I)  (12), 

"we  should  find  the  following  relations  : — 

The  first  derived  polynomial,  that  is,  D^,  is  equal  to 
the  sum  of  the  products  of  the  n  factors  op  (12)  taken 
as   many  times  as  there  are  sets  with  n  —  1  factors  in 

EACH  SET. 

The  SECOND  derived  polynomial,  that  is,  i>2,  IS  EQUAL 
TO  THE  SUM  OF  THE  PRODUCTS  OF  THE  n  FACTORS  OF  (12) 
TAKEN  AS  MANY  TIMES  AS  THERE  ARE  SETS  WITH  71  —  2  FAC- 
TORS IN  EACH  SET. 

The  DERIVED  POLYNOMIAL,  i>„  _  1,  IS  EQUAL  TO  THE  SUM 
OF  THE  n   FACTORS  OF  (12). 

EQUAL  ROOTS. 

2'YO.  If,  in  (6)  and  (t?^),  §  269,  we  make  a  =  b,  the  equations 
become  x^  -\-  px^  -\-  qx  -\-  m  =  (x.  —  o,)^  (x  —  c), 
and  3x^  +  2px  -\-  qz=z  (x  —  a)^  -\-2  (x  —  a)  {x  —  c), 

the  second  members  of  which  arc  each  divisible  by  x  —  a.     Hence, 


EQUAL     R  0  0  T  S.  267 

If  the  equation  a.^  -\-  jpx^  -{-  qx  -\-  m=.^  contains  equal  roots,  the 
equation  and  its  derived  polynomial  will  have  a  common  divisor 
containing  that  root;    and,   conversely, 

If  the  equation  and  its  derived  polynomial  have  a  common 
divisor,  the  equation  has  equal  roots. 

If  we  make  a-=h  =  c,  the  equations  become 

a;'  -f  px^  •}-  qx  -\-  m  =   (x  —  a)^, 
and  3x^  -f-  2j)X  -\-  q  =  ^{x  —  ay, 

the  second  members  of  which  are  each  divisible  by  (x  —  ay. 
Hence, 

If  the  equation  has  three  equal  roots,  the  greatest  common  di- 
visor of  the  equation  and  its  derived  polynomial  contains  two  of 
these  equal  roots. 

And,  conversely,  if  the  greatest  common  divisor  contains  two 
equal  roots,  the  given  equation  contains  three  roots  of  the  same 
value.     And,  generally. 

If   the   equation    x"  -f  px''—'^  +  qx''  —  '^ .  . .  .  -\-  tx  -^  m  =  0 

CONTAINS  EQUAL  ROOTS  TO  THE  NUMBER  OF  S,  THE  EQUATION 
AND  ITS  DERIVED  POLYNOMIAL  HAVE  A  COMMON  DIVISOR  CON- 
TAINING  S  —  1    OF    THESE    EQUAL   ROOTS  ;     or. 

If  the  GREATEST  COMMON  DIVISOR  OF  AN  EQUATION  AND 
ITS  DERIVED  POLYNOMIAL  CONTAINS  S  —  1  EQUAL  ROOTS,  THE 
EQUATION   CONTAINS   S   ROOTS   OF   THE    SAME    VALUE. 

EXAMPLES. 

1.  Find  the  roots  of  the  equation  x^  —  llx^  -f  32:c  —  28  =  0. 

The  FIRST  DERIVATIVE  is  Sx^  —  22x  -f  32  =  0 ;  the  roots  of 
which  are   x  =  2  and   Ls. 

Of  the  factors  x  —  2  and  x  —  \^  the  former  is  that  which 
will  divide  the  given  equation  (^Vide  §256);  .-.  the  roots  are  2,  2, 
and  7. 

And,  in  fact,  (.r  —  2"/  {x  —  7)  =  x^  —  Ux^  -f  32a:  ~  28, 


268        GENERAL  SOLUTION  OF  THE 

2.  Find  the  roots  of  the  equation  x^  —  lOx^  -{-  33x  —  36. 

(Vide  84,  Ex.  8.)  Ans.  3,  3,  and  4. 

3.  Find  the  roots  of  the  equation  x^  —  13x^  +  ^6x  —  80. 

(Vide  84,  Ex.  9.)  Ans.  4,  4,  and  5. 

4.  Find  the  roots  of  the  equation  cc*  —  5^^  +  9x^  —  7x  +  2. 

(Vide  84,  Ex.  6.)         Ans.  1,  1,  1,  and  2. 

5.  Find  the  roots  of  the  equation  x*  —  lOx^  +  37x2  —  60x  + 
36.  (Vide  84,  Ex.  7.)         Ans.  3,  3,  2,  and  2. 

GENERAL  SOLUTION  OF  THE  EQUATION  OF  THE  THIRD  DEGREE. 

flKl,    By  §  268,  any  equation  of  this  degree  may  take  the  form 

x^  •}•  qx  -\-  m  =  0.  (1) 

If  X  =  y  +  r,  we  have  x^  =  ?/^  +  *^  +  ^P'  (.y  +  ^')  J 

Or,  which  is  the  same  thing,  x^  =  ^^  +  /^  +  o^r.x. 

Hence  x'  —  Si/r.x  —  7/^  —  r^  =  0.         (2) 

Therefore  x^  +  g-x  +  ^^^  =  ^'^  —  3^r.x  —  i/^  —  r^.  (3) 

Hence,  by  §  237,   i/^  -{-  r^  =  —  m,  and  3?/7'  =  —  q. 


Whence  y  =^  -  -  +  ^^  +  |^,  and  r=\  ~  ^  -  \-^  +  |^- 

Therefore  x  =;/-f+V?+|  +^/-f-^/f^•  W 
If  we    suppose   this  root  to  be  equal  to  a,  the   factors   of  (1) 

become         x^  '\-  qx  ■{-  m  =  (x  —  ci)  {;x^  +  ox  +  «^  +  2"); 
Therefore  x"^  -\-  ax  -\-  a^  -\-  q  ==  0, 

whence  x  =  ^J  (—  a  db  V^'Sa^  —  4^).  (5) 

Equations  (4)  and  (5)  contain  the  roots  of  (1). 

EXAMPLES. 

What  are  the  roots  of  the  equation  x' —  ISx^  +  lOlx  =  180? 
By  making  the  second  term  disappear,  we  have 

f—7i/  =  6. 
Then  q  =  -^7  and  m  =  —  6. 


EQUATION  OF    THE  THIRD  DEGREE.      269 


By  substituting    these  values  of  q  and    m  in  equation  (4)^  we 


have 


X 


=  ..3+  io;/_3.|_      v>_io^/_3. 


Now,  each   of   these    terms    may  be   expanded    by  the  binomial 
theorem,  and    added    together.      The  terms   involving  j/—  3  will 

cancel,    leaving   x   a    real    quantity.       This    is    called  the    irre-  | 

DUCIBLE    CASE,    and    always    arises    when    t-  "^  97   ^^    ^   negative 

quantity.      Its   occurrence  renders   formula  (4)  entirely  useless  in  | 

practice.     The  roots  of  the  given   equation  are  4,   5,  and  9,  and  ] 
yet   the   formula   will   not    reveal    them   without   the    use    of    an 

infinite  series.  ' 

SI'S.           Numerical  SGlution  of  Cubic  Equations.  ' 

Take  the  equation  | 

x^  -\-  px"^  -\-  qx  =  m.                         (1)  ! 

I 

Find   by  trial  a  number  which,    on  being   substituted   for  x  in  I 

the   given   equation,  will    produce  a  result  less  than  m,  but   such  I 

that  if  it  is  increased   by  uniti/^  and  again  substituted,  the  result  ! 

will  be  greater  than   m.      Let  r  be  such  a  number.     Then,  if  we  ! 

regard  it  temporarily  as  the  exact  root,  we  may  write  | 

i 

f  3  _^  pyi  _^  ^^.  __  ^11 .                            (^2)  I 


Whence 


111 


q  +  p''  +  '''^ 

Having  found  ?*,  denote  the  remaining  figures  of  the  root  by  n, 

wheace 

^  =  /•  +  >J-  (3) 

Substitute  this  value  for  ./:  in  equation  (1),  and  we  have 

(>•  +  if  +  p  {>•  -V  u'y  +  2  ('•  +  :j)  =  '»•        W 

Expand,  and  arrange  in  reference  to  y,  and  we  have 

;/  4.  (3r  +  iOy  +  (3''  +  2i)r  +  g)  y  +  (/'  +  P'-^  +  qr)  =  m.     (5) 

23- 


270  GENERAL     SOLUTION     OF     THE 

3Iake  i^  =  or  -f  /:>,  (j^  =  Sr^  -f  2p^  +  2^? 

and  m^  =  ??i  —  (r^  4-  V'^  ^"  !?'')? 

and  we  have  y^  -}-  p^j/^  +  cj^y  =  ??i^  (6) 

The  first  figure  in  tlie  root  of  equation  (6)  which  may  be 
found  in  precisely  the  same  way  as  in  equation  (1),  is  the  second 
figure  in  the  root  of  (1).  By  repeating  the  process,  the  third, 
fourth,  fifth,  &c.  figures  of  the  root  of  (1)  may  be  found. 

In  applying  the  preceding  principles,  we  proceed  as  folloAvs : — 

1.  Arrange  the  coefiicients  with  their  signs  in  a  line,  and  to 
the  right  of  them  place  the  right-hand  member  of  the  equation. 

2.  Having  found  the  first  figure  of  the  root,  multiply  it  into 
the  first  coefiicient  and  add  the  product  to  the  second  coefficient, 
which  sum  multiply  by  the  same  figure  of  the  root,  and  add  the 
product  to  the  third  coefficient,  multiply  this  sum  by  the  same 
first  figure  and  subtract  the  product  from  the  term  constituting 
the  second  member  of  the  equation. 

The  remainder  is  the  first  dividend. 

3.  Multiply  the  first  coefficient  by  the  same  first  figure  of  the 
root,  and  add  the  product  to  the  last  number  under  the  second 
coefficient;  which  sum  must  be  multiplied  by  the  same  figure, 
and  the  product  added  to  the  last  number  under  the  tJiird  co- 
efficient. 

This  last  sum  is  the  first  tried  dicisor. 

Multiply  the  first  coefficient  by  the  first  figure  of  the  root,  and 
add  the  product  to  the  last  figure  under  the  second  coefficient. 

4.  Divide  the  first  dividend  by  the  first  trial  divisor,  and  the 
(juotient  is  the  second  fi<<jiire  of  the  root.  "With  this  figure  pro- 
ceed exactly  as  with  the  first  figure,  observing  the  rules  fur  deci- 
mals, signs,  &c. 

5.  After  reaching  the  fourth  or  fifth  decimal  place,  four  or 
five  other  places  may  be  found  by  dividing  the  last  dividend  l;y 
the  last  trial  divisor. 


EQUATION     OF    THE    THIRD     DEGREE.  271 
EXAMPLES. 

1.    Given  x^  +  x" -^  x  =  100,  to  find  the  values  of  x. 

Operation. 

Ill  100  I  4.264429973 

5                      21  ^ 

9                   X57  *16 

13.2                 59.64  11.928 

13.4               X62.32  *4.072 

13.66               63.1396  3.788376 

13.72            X63.9628  *.283624 

13.784             64  017936  256071744 

13.788           X64.073088  ^^27552256 

13.7924           64.0786496  25631441984 

13.7928         X64.08412208  ^:a920814016 

13.7932  1281682441 


639131575 

576757098 
"62374477^ 

57675700 

4698768 

4485888 

212880 

192252 


The  numbers  maiked  x  are  divisors;  those  raarkcd  "^  arc  cor- 
responding dividends.  The  reason  for  a  simple  division  after 
reaching  the  fifth  figure  is  apparent. 

The  other  roots  are  now  obtained  as  follows : — 

Divide    .x^  -f  a:^  +  X  —  iOO  =  0  by  x  —  4.264429973,    and  we 

have  X-  +  5.264429973X  =  -  23.449792962^ 

Whence       x  =  -  2.632214986  ±  4.063402165  l^-  1. 


272 


„:^^ 


GEXERAL     SOLUTION     OF     THE 


2.    Find  a  root  of  the  equation  x^  -f  lOx^  +  5x  =  2600. 


0  //n^-<^ 

Wc//  Operation.    ■* 

10     < 

y //5 

2600  11.0067993399 

21 

236 

588 

2596 

32 

4 

43.006 

588.258036 
588.516132 

3.529548216 

43.016 

470451784 

43.0227 

588.54624789 
588.57636427 

411982373523 

43.0234 

58469410477 

43.02419 

588.5802364471 
588.5841086323 

52972221280 

43.02428 

5497189196 

529722 

19996 

17657 

2339 

1765 

574 

522 


X 


52 
45 

7 


Here  we  commence  with  11,  and  the  process  is  precisely  the 
same  as  in  the  preceding  example.  In  the  division  only  the  con- 
stant part  of  the  divisor  need  be  considered,  and  the  correspond- 
ing part  of  the  dividend. 


EQUATION     OF     THE     T II I K  D     DEGREE. 


- 1  o 


3.    Find  .r  in  the  equation  .r^  —  2.v  =  5. 

This  equation  is  the  same  as  .x^  riz  Ox^  —  2x  =  5. 


Operation. 


0 
2 

4 

6.09 

6.18 

6.274 

6.278 

6  2825 

6.2830 

6.28305 

6.28310 


2 

5 

2.094551482 

2 

4 

10 

1.000000 

10.5481 

949329 

11.1043 

50671 

11.129396 

44517584 

11.154508 

6153416 

11.15704925 

6578824625 

11.10079075 

574591375 

11.1011049025 

558055245 

11.1014190575 

16536 

11161 

5375 

4464 

911 

882 

'  ^r':?-''^^^^ 

29 

22 

7 

274  GENERAL     SOLUTION     OF     THE 

4.  Find  x  in  tlic  equation  x^  —  x-  —  15x  =  —  24. 

Oj)eration. 


- 1 

-  15 

-  24     1.7550451874 

0 

-  15 

-  15 

1 

-  14 

—  9 

2.7 

—  12.11 
-  9.73 

-  8.477 

3.4 

-.523 

4.15 

—  9.5225 

-  9.3125 

-  .476125 

4.20 

-  46875 

•   4.255 

-  9.291225 

-  9.269925 

-  46456125 

4.260 

-  418875 

4.26504 

-  9.269754 

-  9.269584 

—  370790175936 

4.26508 

-  48084824 

-  463475 

-  17373 

—    9269 

-  8104 

-  7415 

-  689 

-  648 

-  41 

-  36 

EQUATION  OF  THE  THIRD  DEGREE.      2iO 

5.  Find    the    roots  of    tlic    equation    ;t^  —  1242^=  +  9858x  = 

-  17276. 

Operation. 

1  -1242  9858  -  17276  1 1009  +  200  +  30  +  4  =  123 1. 

_    242  —232142  —232142000 

758  525858  232124724 

1758  917458  183491600 

1958  1349058  ^48023124 

2158  1420698  42620940 

2358  1493238  6002184 

2388  1503046  6002184 
2418 
2448 

2452 
Divide'tliG  glvca  equation  by  x  —  1234,  and  we  have 

x"^  —  8j:  =  14; 

Whence  x  =  9.477225,  or  - 1.477225. 

6.  Given  13..^  +  l^r  -  100|.r  =  -  12A,  to  find  the  values  of  x. 


1o 
O 


Operation. 
—  100.25  —  12.5  :  .125  =  J 


1  (375  „  100.0825  -  10J00825_ 

2.975  _  99.785  -  2.49175 

4.275  -  99.6943  -  L993886 

*        4.535  -  99.5984  ~  .497864 

4  795  —99.5728  —  .497864 

5.055 
5.120 
Divide  the  given  equation  by  .r  —  I,  and  we  have 

13:c2+  2x  ='100; 
Whence  x  =  2.6975,  or  —  2.^514. 

7.    Given  :c'  —  15x'^  +  63x  =  50,  to  find  the  values  of  x. 

Ans.  X  =  1.028039231,  x  =  6.576535,  and  x  =  7.395426. 


276                                H  I  Q  II  E  R     E  Q  U  A  T  I  0  N  S.  i 

i 

8.  Find  a  root  of  the  equation  x^  -f  x^  =  500. 

Ans.  X  =  7.61727975,  &c.  | 

9.  Find  a  root  of  the  equation  x^  -\-  x  =  500.  i 

Ans.  X  =  7.89500828,  &c.  | 

10.  Find  a  root  of  the  equation  x^  +  2x^  +  3x  =  13089030.  , 

Ans.  x=2dD.  ' 

HIGHER   EQUATIONS.  i 

2^3.    The  same  method  may  be  pursued  in  solving  a  numeri-  i 

cal  equation  of  any  degree  whatever. 

1.  Given  a-*  -f-  4x3  ^  3^,2  _|_  2x  =1400,  to  find  x. 


Operation. 

4 

3 

2 

1400  5.216114541,  &c. 

9 

48 

242 

1210 

14 

118 

X834 

=i^l90 

19 

213 

875.568 

175.1136 

24.2 

217. 

84 

X  920.112 

*  14.8864 

24.4 

222 

72 

922.300881 

9.22390881 

24.6 

227. 

64 

X  924.672244 

*  5.66249119 

24.81 

227. 

8881 

926.043446 

5.55626067 

24.82 

228. 

1363 

X  027.415 

•-^  .10623052 

24.83 

228. 

3843  . 

927.43 

•  9274384 

24 

228 

X  927.4 

*  134866 

24 

228. 

92746 

24 

228 

42120 

24 

228 

37098 

24 

228 

* 

5021 

4637 

384 

370 

14 

9 

5 

HIGHER     EQUATIONS.  277 

We  have  first  arranged  the  coefficients  and  second  member  of 
the  equation  in  a  line.  "We  find  by  trial  the  first  figure  of  the 
root  to  be  5.     The  work  then  proceeds  thus : — 

1x5  +  4=9  in  the  first  column,  5  x  9  -f  3  =  48  in  the 
second  column,  48  x  5  +  2  =  242  in  the  third  column,  242  x  5 
=  1210,  the  first  subtrahend. 

1  X  5  +  9  =  14  in  the  first  column,  14  x  5  +  48  =  118  in 
the  second  column,  118  X  5  +  242  =  832,  the  first  trial  divisor. 

1  X  5  +  14  =  19  in  the  first  column,  and  19  X  5  +  118  = 
213  in  the  second  column. 

1  X  5  +  19  =  24. 

We  now  find  the  quotient  of  the  first  dividend  by  the  trial 
divisor  to  be  .2.     Then, 

1  X  .2  +  24  =  24.2  in  first  column,  24.2  x  .2  -f  213  =  217.84 
in  second  column,  217.84  x  .2  +  832  =  875.568  in  third  column, 
and  875.568  x  -2  =  175.1136,  the  second  subtrahend. 

Then  1  x  .2  +  24.2  =  24.4  in  first  column,  24.4  x  .2  + 
217.84  ==  222.72. 

222.72  X  .2  +  875.568  =  920.112,  the  second  trial  divisor. 

1  X  .2  +  24.4  =  24.6  in  first  column,  24.6  x  .2  +  222.72  = 
227.64. 

1  X  .2  +  24.6  =  24.8. 

We  now  find  the  quotient  of  the  dividend  by  the  second  trial 
divisor  to  be  .01.     With  this  proceed  exactly  as  with  the  others. 

As  the  decimals  on  the  right  do  not  afiect  the  result  when  a 
limited  number  of  figures  is  sought,  we  may  disregard  them  in 
the  calculation. 

On  arriving  at  the  fourth  decimal  place,  we  simply  divide  for 
the  others. 

To  this  equation  we  may  find  another  root,  thus : — 

By  trial,  —  7  is  found  to  be  the  first  figure,  when  we  proceed 
as  follows: — 


278  II  I  G  II  E  R     EQUATION  S. 


4 

3 

2 

1400  j  -  7.26948592 

-  3 

24 

—  168 

1162 

-10 

94 

-824 

238 

-17 

213 

-867.568 

173.5136 

—  24.2 

217.84 

-912.112 

64.4864 

-24.4 

222.72 

-925.859896 

55.5515937 

-24.6 

227.64 

-939.697504 

8.9348063 

-  24.86 

229.1316 

—  941.78866 

8.4760979 

-  24.92 

230.6268 

-943.89185 

4587084 

—  24.98 

232.1256 

—  943.9849 

3775939 

—  25.049 

232.3510 

-944.078 

81114 

—  25.058 

232.5765 

75525 

-  25.067 

232.8021 

5589 

—  25.0764 

232.81 

4720 

-  25,0768 

232 

869  , 

846        f 
23 

_i8         : 

6 

The  work  in  this  example  will  be  readily  followed.  The  other 
two  roots  might  be  found  by  dividing  the  given  equation  by 
X  +  7.26948592,  and  then  hj  x  —  5.216114541,  thus  reducing 
it  to  a  quadratic  equation,  to  be  solved  in  the  usual  way. 

2.  Find  one  root  of  the  equation  a;*  +  4x^ —  4^^  —  II.1;  =  —  4. 

Alls.   X  =  1.63691356,  kc 

3.  Find  one  root  of  the  equation  a"  —  Sx^  +  75x  =  10000. 

Ans.  0.88600270094,  &c. 


QUESTIONS     FOR     EXAMINATION.  2  <  'J 


QUESTIONS   FOR  EXAMINATION. 

1.  What  is  the  numerical  vaUie  of  —^ ^   ,    -.   +         ,    ^ 

when  ?i  =  1  and  x  =  1,  2,  3,  4,  5,  &c.  ? 


X"  —  1 

2.  What  is  the  numerical  value  of  rt~-(  1  +  -  |     \^-hen  o  =  1,  2, 

3,  4,  ^,  J,  i,  &c.,  and  x  =  1,  2,  3,  4,  ^  ^  i,  &c.  r' 

3.  What  is  the  value  of  Ji  1  —  ^  )'  when  a  =  j^T  and  a;  =  0  ? 
when  «  =  3  and  x  =  1?  «  :=  6  and  x  =  o? 

(rv.4  \    1 
1  +  ^  )'  when  cj  =  2  and  .x  =  0  ? 

when  rt  =  2  and  a;  =  j/  (35  ? 

r     »  1  w        1        x2  +  a;  —  20  x2  +  2x  —  15 

5.  Add  to":ether  — =—  and  — t^t^. 

°  x2  —  7x  +  12  x2  +  a;  —  20 

^    ^  ct;2  +  .-c  —  20       ,     ^2  _(_  2x  —  15 

^-   ^^'«^  x--7a;  +  12  '^^^  .- +  .  -  20  " 

x2  4-  ic  —  20   ,      a;2  +  2x  —  15 

7.  Multiply  -T-^ — ;-^r^  by  -;r^7 --. 

^  -^  x^  —  7ic  -I-  12    -^    x^  +  X  —  20 

^    ^.  . ,     x^-{-x  —  20  ,     5c2  +  2x  —  15 

8.  Divide  — — — .rrr  by  -^—-. wtt- 

9.  Find  the  value  of  x  in  the  equation 

29 ^  ^4  —        —  ^-02. 

10.  Find  the  value  of  x  in  the  equation 

5x  —  1       3.r  —  2       llx  —  3        13x               8x  —  2 
2o  +  -^^—  +  —^  .       ^2        =  ~3"  ~  ^ 7~~' 

—  -         ^~j  «  i^   ~i~    UX   '~~   ox  ex    ~"~    C(i  n       1 

11.  Griven 5 = ,  to  find  x. 

a  —  6  c 

n"^  -L  nh   -i-   h^ 

12.  Given  a' +  h' -\-  a'x  -  l^x  =  .       7^     ,  to  find  x. 

cr  —  0^ 


280  QUESTiojrs  fou  examination. 


13.  Given  :^  -f  f ,  =  15  and  7^  -I'  y  =  1^^'''»  to  fin'i  ^  and  y. 
4i  41  "^ 

n    r--       3      4      ,       ,  ^;  ,   10      ^        ,.   ,  '' 

li.  Uriven  -  -f  -  =  4  and       -^ =  9,  to  find  x  and  y. 

X       y  X         ;y  ^  "^ 

y1y//j.   z  =  1  ^.  y  ==  2. 

1.0.   Giv'rn   X  +  -  =  /y  and  J  -f  y  =  d.  to  find  :/:  and  y. 
<j.  c 

.                oZ/c  —  cd            and  —  ah 
Aruf.  X  = =— ,  y  = =— . 

OA  —  1  o.r,  —  1 

/  X  -\-  aJ y  -\-  z  -\-  w)  =  ra, ' 

.y  +  '>  ''^^  4-  .y  4-  '^v  =  '^h  I 

IG.   Given    '        ,       ,         '  y    U>  find  y:,  y,  z,  and  ip. 

(  v;  +  d  (x  +y  -h   z)  =  y,   J 

.  m  ^^/       1  —  a       \  —  n       1  —  c       1  —  a 

17.  Given  —1^—^  =>  !t^^,  to  find  the  value  of  :/:.     yl/i«.  :/:  =  5.      1 

l'^    Given  ^'  „       ,  =    ^~ -.  to  find  the  values  of  x. 

2x*  —  6       X*  4-  a  

vln«.  X  =  =t  ya  4-  ^• 
J'^  Givf;n  x''  —  :/,  —  '/'),  U)  find  the  valuea  of  x.  1 

AriM.  X  =  H   and    —  7.        j 

20.  Given  — ~ ^ —  =  7,  to  find  the  valuen  of   x. 

X  x^ 

Ana.   X  =  'J   or    —  ^.       ' 

2J.    Giv<;n    lOx  —  x^  =  05,   to  find   tlje  values  of   x.  ' 

yfw;?.  X  =  8  db  1/^^.        ■ 

..o    ^..  -       1017x  2071^-         /.,,./.  J 

22.   (fi\'<:f)   '//  —        ,     ==  —         ,     ,  to  ijn'i   til';    vaJuCrt  of    x.  1 

2i  2J    '  ; 

Am.  x  =  00.72  or   10.27.       1 

1 

2'i.   GIvf;n   X  —  x^  =  20,  to  find  tlio  valu^iH  of  x.       vl//,;,'.   x  =  25.        i 

21    Given  (x  4  \()/  —  ( j.  4  10/  =  2,  to  find  x.     yl««.  x  =  6.       j 


Q  V  i;  >>  T  [  0  X  ;>    r  o  u    i:  x  a  m  i  n  a  t  i  o  :;. 


281 


25.  Given  (7.r=  —  .'•  H-  !>'  —  ("••'''  —  •'"  +  1)  =  '^'^-^  ^'^  ^''^'^  '^^'^ 
values  of   .r.  A 11^.   .r  =  /j  (1  ±  1  '^r^f),  .r  =  'J   or   —  'j-'- 

20.   ("livea  .<"  +  2.;- =  2:"!,  io  ilnd  ,r. 

.l;rs-.   .r  =  3.808070r>   and  .r  -=  —  r).S0S07!^r). 


27.  (Uvon     


//.s.  .(•  =  5, 


1  — 


.  -I-  I 


2.r// 


28.   Given 


.M-.V--:^y 


.'^' +.'/    ^  1  .,„a  7.,.,,  :=^  -S,  lo  1111,1  .r  a.ul  y. 
x^  +  y'  '' 

An..   .r  =  2,//  =  2. 


2!>.   (Jlveu   ■--    ..•''^-_:i-21:j^  ==  (/   ami   .-y  =  A,  (o   lin.d   .r   ar.d  //. 
X  +  ,y       a;  —  y 

X  —  11       X  -}-  ,y 

,!;,.•.    ,7=   \    [^=ir^     (.'/;4-26=Fl/(7/>  — 2A], 
.,.  ^  ),    [riz  ^    ,//,  _|.  •_'/,  =h  1   V//>  —  -/.]. 

^JO.   G  ivoii  .r  -f-  //  -f  1 "".»;  +  //  =  0  aii.l  .r'  +  if-  =  1 0,  to  liiul  ,r  ami//. 

J.;n.   .(•  =  -  or  1,  or   U  ±  ^1  '—Ol, 
//  =  1  or  :l,  or  1]  qz  jl    —  1)1. 

ai.   Given       J  ^  ^.    —      „'  ,  .   =  l.J\,  U)  lind   .r. 


4/?.s-.    .r  =  8,  or 


1  t 


o2.   Given 


1.  ~.r-  f  1  ^  •-'"  *"   ^'"'^  •'' 


.u/s.   .r 


=  ±3,  or  .r  =  ii=  \\    —  1 


33.  One   root  of  .<^  —  in.Vr- +  4(>^r  =  2i)    is  8:    what  arc   the 
other  roots?  J/zs.   .r  =  5  and  .r  =  ^. 

84.   One  root  of  ./•'  -f  Iy\ y^  —  \~^x  =  —  1    Is  —  (> :    what  are  the 
other  roots?  yi;?.9.   \  and   \ 

O  1 


282  Q  U  E  iS  T  I  0  N  S     F  0  K     E  X  A  M  I  N  A  T  I  0  N. 

35.  Has  the  equation  ;/.^  -}-  2^.'-^  —  15x  =  36  eqnal  roots  ?  If  so, 
find  all  the  roots. 

Ans.  It  has;    and  —  3^  —  3^  and  —  4  are  the  roots. 

36.  What  are  the  roots  of  the  equation  x^  —  20^2  -f  142./2  — 
420x  =  -  441  ?  Ans.  7,  7,  3,  and  3. 

37.  Find  one  value  of  x  in  the  equation  x^  —  2x  =  50. 

Ans.  3.8648854. 

38.  If  a  certain  number  is  divided  by  7,  and  if,  then,  the  quo- 
tient is  taken  from  the  sum  of  the  dividend  and  divisor,  the  re- 
mainder will  be  73  :   what  is  the  number?  Ans.  77. 

39.  To  find  three  numbers  in  arithmetical  progression,  of  which 
the  first  is  to  the  third  as  5  to  9,  and  the  sum  of  all  three  is  63. 

Ans.  15,  21,  27. 

40.  A  sets  out  from  C  toward  D,  and  travels  8  miles  a  day. 
After  he  had  gone  27  miles,  B  set  out  from  D  toward  C,  and  goes 
every  day  379^^^  ^^  ^^^®  whole  journey,  and  after  he  had  traveled 
as  many  days  as  he  goes  miles  in  one  day,  he  met  A.  Required 
the  distance  of  tlie  place  C  from  D.  Ans.   180  or  60  miles. 

41.  Two  post-boys,  A  and  B,  set  out  at  the  same  time  from 
two  cities,  500  miles  apart,  in  order  to  meet  each  other.  A  rides 
60  miles  the  first  day,  55  the  second,  50  the  third,  and  so  on, 
decreasing  5  miles  every  day.  B  goes  40  miles  the  first  day, 
45  the  second,  50  the  third,  and  so  on,  increasing  5  miles  every 
day.     In  what  number  of  days  will  they  meet?     Ans.  In  5  days. 

42.  A  tree,  100  feet  high,  stands  just  at  the  water-line  on  the 
bank  of  a  river  200  feet  wide.  The  tree  broke  in  a  gale  of  wiud, 
and  the  upper  part  was  found  to  point  exactly  to  the  water-line 
of  the  opposite  bank,  the  top  being  within  20  feet  of  the  surface 
of  the  water.     How  high  from  the  ground  did  the  tree  break  ? 

Ans.  30.472  feet. 


TABLE    OF    SOUAIIK    J'.OOTS. 


r: 


No.   fc?q\<u:"'j  K(.)ot. 


0 

6 

7 
S 

Q 
lO 
)  I 
12 

i3 

i5 
i6 

17 
l« 

'9 
20 
21 

22 
2? 
2-t 
25 
26 
27 
28 
29 

3o 
3i 

32 

33 
34 
35 
36 

37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 

49 
5o 
5i 

52 

53 
54 
55 
56 

57 
58 

59 
60 


0000000 
4 142 1 36 
732o5o8 
0  00000 
2060680 
4494897 
6437313 
8284271 
0000000 
1622777 
3166248 
4641016 
6o555i3 
7416574 
8729833 
0000000 
i23io56 
2426407 
3588989 
4721360 
5S25757 
6904158 
79583 1 5 
8989795 
0000000 
0990  up 
1961524 
2915026 
385 1648 
4772206 
5677644 
656854'i 
7445626 
8309519 
9160798 
oooouoo 
0827625 
1644140 
2449980 
3245553 
4o3i24'2 
4S07407 
5574385 
6332496 
70S2039 
7S23300 
8556546 
9282032 
0000000 
0710678 
14 14284 
2 1 1 1026 
2801099 
3484692 
4161985 
4833148 
5498344 
61 57731 
681 1457 
7459667 


No. 
61 

62 

63 
64 
65 
66 
67 
68 
69 
70 
71 
72 
73 
74 
75 
76 
77 
73 

79 

80 

81 

82 

83 

i)4' 

85 

86 

«7 
88 
89 
90 
91 
92 
93 
94 
95 
96 

97 
98 

99 
100 

101 

102 
io3 
1 04 

iG5 

106 

107 
108 

109 
I  to 
1 1 1 

I  12 

113 

114 

ii5 

n6 

H7 
118 

119 

120 


Squ;ire  Knot. 

7' 8102497 

7 

8740079 

7 

9372539 

8 

0000000 

8 

0622077 

8 

1240384 

8 

1853528 

8 

24621 1 3 

8 

3066239 

8 

3666oo3 

8 

4261498 

8 

4852814 

8 

5440037 

8 

6023253 

8 

6602540 

8 

7'77979 

8 

7749644 

8 

83 ! 7609 

8 

8881944 

8 

9442719 

9 

0000000 

9 

o55385i 

9 

1 104336 

9 

i65i5i4 

9 

2195445 

9 

2735t85 

9 

3273791 

9 

3So83i5 

9 

4339811 

9 

4868330 

9 

5393920 

9 

0916630 

9 

6436008 

9 

6953597 

9 

7467943 

9 

7979090 

9 

8488578 

9 

8994949 

9 

9498744 

10 

0000000 

10 

0498706 

10 

0995049 

10 

1 4889 16 

10 

1980390 

10 

5469508 

10 

2956301 

10 

3440804 

10 

39230-48 

10 

44o3o6o 

10 

4880885 

10 

5356538 

10 

5830002 

10 

6301458 

10 

67707S3 

10 

7238o53 

10 

7703296 

10 

8i66538 

10 

8627800 

10 

90S7121 

10 

9544012 

rsi.>.   j  !S4U;ii-e  Itoot. 


L  J14JI ,  VaMaWJUi^ESrU.!^ 


',*g.wt:mi'..ja*.p;T^g».--  <.JttJ-gTgL,*!iJgg. 


121 

II  • 

122 

1 1  • 

123 

1 1  • 

124 

1 1  • 

120 

1 1  • 

126 

11  • 

127 

1 1  • 

128 

1 1  • 

129 

II  • 

i3o 

1 1  • 

i3i 

ii> 

l32 

1 1  • 

i33 

II  • 

1 34 

1 1  • 

i35 

i  { • 

i36 

II- 

137 

II  • 

i38 

II  • 

,39 

11  • 

140 

1 1  • 

141 

1 1  • 

142 

11  • 

143 

II  • 

144 

12- 

145 

12- 

146 

12- 

147 

12- 

148 

12- 

149 

12- 

!0O 

12- 

i5i 

12- 

l52 

12- 

!53 

12- 

104 

12- 

i55 

12- 

1 56 

12- 

i57 

12- 

i58 

12- 

159 

12- 

i6o 

12- 

161 

12- 

162 

12- 

16J 

12- 

164 

12- 

i65 

12- 

166 

12- 

167 

12- 

168 

12- 

169 

i3- 

170 

i3- 

171 

i3- 

172 

i3- 

173 

i3. 

174 

i3- 

175 

i3- 

176 

i3- 

177 

i3- 

178 

i3- 

179 

i3- 

180 

i3- 

0000000 
0453610 
■0905360 
1355387 
1803399 

2249722 

2694277 

3  70S  5 

•3578167 

4017543 

■445523i 

•4891253 

532  5626 

5708369 

■6189500 

■6619033 

•7046999 

■7473444 

•7898261 

•8321096 

•8743421 

•9163753 

■9582607 

•0000000 

•0410946 

•o83o46o 

1243557 

i65525i 

2o65o56 

2474487 

•2882057 

•3288280 

■3693169 

1096736 

4498996 

4899960 

■5299641 

56980O1 

•6095202 

■6491 106 

•6885775 

■7279221 

•7671453 

•8062485 

■8452326 

•8840987 

•9228480 

■9614814 

■0000000 

•0384048 

■0766968 

1148770 

1529464 

1909060 

2287566 

2664992 

•3o4i347 

•3416641 

•3790S82 

•4I6407Q 


JJMiJUldaMg'ITIJJI'S-'igg 


A    TABLE    OF     LOCiAlllTIiMS     FKOil     1     10     10,000. 


N. 

0 

I 

2 

3 

4 

5  j  6  |-  7  1  8 

9  i  D. 

100 

000000 

0434 

0S68 

1 3oi 

1734 

2166  2398  3029  3461 

3891;  432 

!0J 

432  1 

4731 

5i8i 

5609 

6o38 

6466'  6894'  732 1  j  7748 

8174'  428 

I02 

8600 

0026 
3239 

945 1 

9876 

e3oo 

•724  1147  1570  1993 

24 1 5 

424 

io3 

012837 

368o 

4100 

4521 

4940|  536o;  6779 

6197 

66}  6 

419 

104 

7033 

7431 

7868 

8284 

8700 

91 16  9532I  9947 

•36j 

*773 

416 

io5 

021 189 

i6o3 

2016 

2428 

2841 

3232  i  3664  4075 

44S6 

4896 

412 

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8944'  9 '06 

9268 

9429 

9591 

162 

269 

9752 

9gi4 

••73 

8236 

•398 

•559  ^720 

•881 

1042 

I203 

161 

270 

43 1 364 

i525 

1 685 

1846 

2007 

2167  2328 

2488 

2649 

2809 

161 

271 

2969 

3i3o 

3290 

345o 

36io 

3770  3(^30 

4090 

4249 

4409 

160 

272 

4569 

4729 

4888 

5o48 

5207! 

5367  5d26 

5685 

5844 

6004 

1 59 

273 

6i63 

6322 

6481 

6640 

6798, 

695/  71 16 

72-'5 

7433 

7592 

159 

274 

775i 

7909 

8067 

82  26 

8384! 

8542 

8701 

8859 

9017 

9175 

1 58 

275 

9333 

9491 

9648 

9806 

9964 

•122 

•279 

•437 

•594 

•752 

1 58 

276 

440909 

1066 

1224 

i38i 

i538 

1695 

i852 

2009 

2166' 

2323 

1 57 

277 

2480 

2637 

2793 

2950 
43131 

3 1 06 

3263  3419 

3576 

3732: 

3SS9 

1 57 

278 

4045 

4201 

4357 

4669' 

4825  4981 

5i37 

5293 

5449 

1 56 

279 

56o4 

5760 

5915 

6071  62261  63821  6537 

6(592 

6848 

7003 

i55 

N. 

0 

I  1 

2  1 

3    4  1  5    6 

7  1 

3 

9_ 

T>,  j 

A    TABLE    OF     I.OC;  A  HI  T  iIM3    FflOM     1     TO     10.000. 


"n:" 

0 

I 

2 

1  3 

i  ' 

1  ' 

1  6  i  7 

8  j  9  1  D. 

'280" 

447158 

73 1 3 

7463 

7623 

:   77  73 

7933 

So38;  8242!  83,77 i  8552 

1 55 

23 1 

8700 

8861 

9015 

9170 

:  9324 

9^78 

9633;  9787!  99  i I 

•»95 

1 54 

282 

450249 

04:3 

0)57 

07  i  1 

!  o355 

.,   1018 

1172;  1 326 

1479 

1 633 

1 34 

2  S3 

n86 

I9=i0 

2093 

2247 

i  2400 

2)')3 

2706;  2339 

3oi2 

{  3i65 

1 53 

2S4 

33 18 

347  1 

362  \ 

3777 

1  3930 

4082 

4235;  4337 

4540 

;  4692 

1 53 

285 

4845 

4997 

5i5o 

53o2 

'■  5454 

56o6 

5753:  5910 

6062 

6214 

132 

2S6 

6366 

65 1 8 

6670 

'  6321 

i  6973 

7125 

72761  7428 

7379 

7731 

l52 

i  2^7 
1    ,  ' 

7S82 

8o33 

81S4 

8336 

8487 

8633 

8789'  8940 

0091 

9242 

i5i 

2VS 

9392 

9543 

9604 

9845 

;  9995 

•i46 

•296  •447 

•597 

•748 

i5i 

2 '^9 

460S98 

1048 

1198 

1 3 ',8 

1499 

i6j9 

f799'  1918 

2098 

2248 

i5o 

29) 

462398 

2548 

2697 

2347 

2997 

3 146 

3296,  34 P 

3594 

3744 

i5o 

291 

3393 

4042 

4191 

43  io 

4490 

4639 

4788,  4936 

5o85 

5234 

149 

1  292 

5383 

5532 

563o 

5329 

5977 

6126 

6274  6423 

6571 

6719 

140 

293 

6868 

7016 

7!  64 

73i2 

7460 

7608 

7756;  7904 

8o52 

8200 

148 

294 

8347 

8493 

8643 

8790 

8933 

■  9035 

9233  9.330 

9527 

9675 

148 

293 

9822 

9969 

•116 

•263 

•4!0 

•537 

•704;  ®35i 

•998 

1145 

147 

296 

471292 

1433 

1 535 

1732 

1878 

2025 

2171'  23 1 8 

2464 

2610 

146 

297 

2756 

2903 

3  049 

319') 

3341 

3487 

3633;  3779 

3925 

4071 

146 

298 

4216 

4362 

45o3 

4653 

4799 

4944 

5090  5235 

533i 

5526 

146 

299 

5671 

58i6 

5962 

6107 

6252 

6397 

6542  6687 

683 2 i  6976 

145 

3oo 

477121 

7266 

74 1 1 

7535 

7700 

7844 

79^^.9;  8r33 

8278,  8422 

145 

3oi 

8566 

8711 

8355 

8909 

9143 

9287 

9 '.3 1  9575 

9719 

9863 

144 

302 

480007 

Ol5£ 

0294 

043  3 

o5S2 

0723 

0869  10 1 2 

1 1 56 

1299 

144 

3o3 

1443 

1 586 

'729 

1872 

2016 

2!  59 

23o2  2443 

2583 

273. 

143 

3  04 

2S74 

3oi6 

3i59 

33o2 

3445 

3537 

3730;  3372 

40  i  5 

4r57 

143 

3o5 

43oo 

4442 

4585 

4727 

4869 

Soil 

5i53|  5295 

5437 

5579 

142 

3o6 

5721 

5863 

6oo5 

6t47 

628q 

643o 

6572!  6714 

6855 

6997 

142 

3o7 

7t38 

7280 

7421 

7563 

7704 

7845 

7986;  8127 

8269 

8410 

141 

3o3 

855 1 

8692 

8333 

S974 

9114 

9255 

9396'  9537 

9677 

9818 

141 

3  09 

9958 

••99 

•239 

•33o 

•520 

•661 

•3oi 

•941 

1081 

1222 

140 

3io 

491362 

l502 

1642 

1782 

1922 

2062 

2201 

2341 

2481 

2621 

140 

3ii 

2760 

2900 

3o4o 

3179 

33 19 

3453 

33o7 

3737 

3376 

4oi3 

139 

3l2 

4i55 

4294 

4433 

4072 

4711 

435o 

49S9  5 1 23 

5267 

5406 

139 

3i3 

5544 

5683 

5822 

6960 

6099 

6238 

6376,  65i5 

6653 

6791 

i39 

3 14 

6q3o 

7058 

7206 

7344 

74S3 

7621 

7759 

7897 

8o35 

8173 

1 38 

3i5 

83ii 

8448 

8586 

8724 

8862 

8999 

9137 

0275 

9412 

9550 

i38 

3i6 

.  9687 

9324 

9962 

**99 

•236 

*374 

•5u 

0548 

•783 

•922 

137 

3i7 

Doio59 

1196 

i333 

1470 

1607 

1744 

1880 

2017 

21 54 

2291 

137 

3i8 

2427 

2  564 

2700 

2337 

2973 

3109 

3246 

3382 

35i8 

3655 

i36 

319 

3791 

3927 

4o63 

4199 

4335 

4471 

4607 

4743 

4873 

5oi4 

1 36 

320 

5o5i5o 

5286 

5421 

5557 

5693 

5828 

5964  6099 

6234 

6870 

1 36 

321 

65o5 

6640 

6776 

6911 

7046 

7.S, 

73i6:  745i 

7536 

7721 

i35 

322 

7856 

7991 

8126 

8260 

8395 

S33o 

8664|  8799 

8934 

9068 

i35 

323 

9203 

9337 

9i7i 

9606 

9740 

9'^74 

•••9  •143 

•277- 

•411 

i34 

324 

516545 

0679 

08 1 3 

09 17 

1081 

I2l5 

1349-  1482 

1616 

i75o 

!34 

323 

1 833 

20[7 

2l5l 

2284 

2418 

2  35  r 

2684 

2818 

2951 

3084 

i33 

326 

3218 

335i 

3484 

3617 

3750 

3383 

4016 

4149 

4282 

4414 

i33 

327 

4548 

463 1 

48 1 3 

4916 

5o79 

521  I 

5344 

5476 

5609 

5741 

]33 

323 

5874 

6006 

6139 

6271 

6403 

6333 

6668 

6800 

6932 

7064 

1 32 

3^9 

7196 

7328 

7460 

7592 

7724 

7355 

7987 

8119 

825i 

8382! 

l32 

33o 

5!85i4 

8646 

8777 

8909 

9040 

9171 

93  o3 

9434 

9566 

96971 

i3i 

33 1 

9828 

9959 

••90 

®22I 

•353 

•48' 

•6i5 

•743; 

•876 

1007; 

i3i 

332 

D2II3S 

1269 

1 400 

i53o 

1661 

1792 

192-2 

2d53; 

2i83 

23i4l 

i3i 

333 

2444 

2575 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

36i6; 

i3o 

334 

3746 

3876 

4006 

41 36 

4266 

4396 

4526 

4636 

4785 

4915; 

i3o 

335 

5o45 

5i74 

53o4 

5434 

5563 

5693 

5822 

5951 

60S  I 

6210' 

129 

336 

6339 

6469 

6593 

6727 

6856 

6935 

7114 

7243] 

7372 

75oi| 

129 

337 

7630 

7759 

7888 

8016 

8145 

8274 

8402 

853 1 ! 

8660 

8788 

120 

338 

8917 

9045 

9'74 

9302 

943o 

9559 

9687 

93,5| 

9943 

••72 

12B 

339 

N. 

530200 

o328 

0456 

o584 

0712 

0840 

0968 

1096I 

1223 

1 

i35i: 

128 

0 

I 

3 

4 

5 

6 

7  1  8  1 

9  ! 

8 

A  T 

\j3LE 

OF  LOG  a; 

:r;ii:.3 

5  Fli 

JM  1 

TO 

10,000. 

"nT" 

0    1   I  j   2 

3 

4 

5  (  6 

7 

8 

9 

■—    1 

340 

531479  1607*  1/34 

i862 

1990 

2117,  2245 

2372 

25oo 

2627 

128 

341 

2754 

28821  3009 

3x36 

3264 

3391I  35i8 

3645 

3772 

3899 

127 

342 

4026 

41 53  4280 

4407 

4534 

4661!  4787 

4914 

5o4i 

5167 

127 

343 

0294 

5421  5547 

5674 

58oo 

5927  6o53 

6180 

63o6 

6432 

126 

344 

6558 

6685  6bii 

6937 

7063 

7189 

73i5 
8574 

7441 

7567 

7693 

126 

345 

7819 

7945  8071 

8197 

8322 

8448 

8699 

8823 

8951 

126 

346 

9070 

9202  9327 

9452 

9578 

9703 

9829 

9954 

•*79 

®204 

123 

347 

540029 

0455  o58o 

0705 

o83o 

0955 

1080 

12o5 

i33o 

1454 

123 

34« 

1 579 

1704  1829 

1953 

2078 

220} 

2327 

2452 

2576 

2701 

123 

349 

2825 

2950  3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

124 

35o 

544068 

4192  43i6 

4440 

4564 

4688 

4812 

4936 

5o6o 

5i83 

124 

35i 

5307 

543 1  5555 

5678 

58o2 

5925 

6049 

6172 

6296 

6419 

124 

3D2 

6543 

6t)66  6789 

6913 

7o36 

7159'  7282 

74o5 

7529 

7652 

123 

353 

7775 

789S  8021 

8144 

8267 

8389 

85i2 

8635 

8758 

8881 

123 

354 

9003 

9126  9249 

9371 

9494 

9616 

9739 

9S61 

9984 

•106 

123 

355 

550228 

o35i 

0473 

0095 

0717 

0840 

0962 

1084 

1206 

1 328 

122 

356 

1430 

1572 

1694 

1816 

1938 

2060 

2181 

23o3 

2425 

2547 

122 

357 

266S 

2790 

2911 

3o33 

3i55 

3276 

3398 

35i9 

3640 

3762 

121 

358 

3883 

4004  4126 

4247 

4368 

4489 

4610 

4731 

4832 

4973 

121 

359 

5094 

52 1 5  5336 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

121 

36o 

5563o3 

6423  6544 

6664 

6785 

600  5 

7026 

7146 

7267 

7387 

120 

36i 

7007 

7627  7748 

7868 

7988 

8ioS 

8228 

8349 

8469 

8589 

120 

362 

8709 

8829  8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

120 

363 

9907 

«e26 

«i46 

«265 

•385 

®5o4 

•624 

•743 

0863 

'982 

119 

364 

56i 101 

I22I 

1 340 

1459 

1578 

1698 

1817 

1936 

2o55 

2174 

119 

365 

2293 

2412 

253i 

2600 

2769 

2887 

3  006 

3125 

3244 

3362 

119 

366 

3481 

36oo 

3718 

3837 

3953 

4074 

4192 

43ii 

4429 

4548 

119 

367 

4666 

47f^4 

4903 

0021 

5 139 

0257 

5376 

5494 

56 1 2 

5730 

118 

368 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

iiS 

369 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7S49 

7967 

8084 

118 

370 

568202 

83 1 9 

8436 

8o54 

8671 

8788 

8905 

9023 

9140 

9257 

117 

371 

9374 

9491 

9608 

9725 

9842 

9959 

««76 

•193 

e309 

•426 

117 

^72 

570543 

0660  0776 

0093 

1010 

1 1 26 

1243 

1339 

1476 

1592 

117 

373 

1709 

1825 

1942 

2o58 

217^ 

2291 

2407 

2523 

2639 

2755 

116 

374 

2872 

29S8 

3 1 04 

3220 

3336 

3452 

3568 

3684 

3Soo 

3915 

116 

375 

4o3i 

4147 

4263 

4379 

4494 

4610 

472-6 

4841 

49^7 

5072 

I!6 

376 

5i88 

53o3  5419 

5534 

565o 

5765 

588o 

0996 

61 1 1 

6226 

ii5 

377 

6341 

6457 

6J72 

6687 

6002 

6917 

7032 

7147 

7262 

7377 

ii5 

37b 

7492 

7607 

7722 

7830 

7931 

8066 

8181 

8295 

8410 

8525 

ii5 

379 

8639 

8754 

8868 

8983 

Q007 

9212 

9326 

9441 

9555 

9669 

114 

3bo 

579784 

9898 

e«i2 

®I26 

^241 

•355 

•469 

•583 

•697 

•811 

114 

38i 

580925 

1049 

11 53 

1267 

i38i 

1495 

1608 

1722 

1 836 

igSo 

114 

382 

2o63 

2177 

2291 

2404 

25i8 

263 1 

2745 

2858 

2972 

3o85 

114 

383 

3199 

33 12  3426 

3539 

3652 

3765 

3879 

3972 

4io5 

4218 

ii3 

384 

433 1 

4444  4557 

4670 

4783 

4896 

5009 

3122 

5235 

5348 

ii3 

385 

5461 

5574  5686 

5799 

5912 

6024 

6i37 

6230 

6362 

6475 

ii3 

386 

6587 

6700 

6812 

6925 

7037 

7149'  7262 

7374 

74S6 

7599 

112 

387 

7711 

7823 

7935 

8047 

8160 

8272:  8334 

8496 

8608 

8720 

112 

388 

8832 

8944 

9o56 

9167 

9279 

9391 

95o3 

9615 

9726 

9838 

112 

389 

9950 

««6i 

•173 

•284 

•396 

»5o7 

•619 

•730 

•842 

•953 

112 

390 

591065 

1 176 

1287 

1399 

i5io 

1621 

1732 

1843 

1955 

2066 

1 1 1 

391 

2177 

2288 

2399 

25lO 

2621 

2732 

2843 

2954 

3o64 

3175 

III 

392 

3286 

3397!  35o8 

36 18 

3729 

3840 

3930 

4061 

4171 

4282 

III 

3c;3 

4393,  45o3|  4614 

472-1 

4834 

4945 

5o55 

5i65 

5-276 

5386 

no 

394 

5496 

56o6 

5717 

5827 

5937 

6047 

6.57 

6267 

6377 

6487 

no 

395 

6597'  6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

no 

396 

7695  7805 

7914 

8024 

8i34 

8243 

8353 

8462 

8372 

8681 

no 

397 

8791:  8900 

9009 

9119 

9228 

9337 

9446 

9556 

9663 

9774 

109 

398 

9S83.  9992 

•lOI 

•210 

•319 

•428:  •537 

•646 

•755 

•864 

109 

399 

600973  1082 

1191 

1299 

1408 

i5i7i  1625 

1734 

1843 

1951 

109 

N- 



0   1  I  ,2 

3 

4 

5  i  6 

7 

8 

9 

T). 

A    TABLE    OF    LOGARITHMS    FROM    1    TO    10,000. 


N.  1 

0 

I  1 

2 

2277 

3 

4  j  5 

6 

7 

8 

9  1  I>- 

400  1 

602060 

2169' 

2386! 

2494'  2603' 

2711! 

2819 

2928 

3o36  108 

401 

3i44 

3253 

336i 

3469' 

3577!  36861 
4658'  4766I 

3794 

3902 

4010' 

4118 

108 

402 

4226I 

4334' 

4442 

455o 

4874 

4982 

5089 

0197 

108 

4o3 

53o5 

54i3 

5521 

5628 

5736!  58441 

5951 

6059 

6166 

6274 

108 

404 

638i 

6489' 

6596 

6704 

681 1 

6919I 

7026 

7133 

7241 

7348 

107 

40  5 

7455 

7562 

7669 
8740 

7777 

7884 

7991 

8098 

8205 

83i2 

8419 

107 

406 

8526 

8633 

8847 

8954 

9061 

9167 

9274 

9381 

9488 

107 

407 

9594 

9701 

9808 

9914 

••21 

•128 

•234 

•341 

•447 

•554 

107 

408 

610660 

0767 

0873 

0979 

1086 

II92 

1298 

i4o5 

i5ii 

1617 

106 

409 

1723 

1829 

1936 

2042 

2148 

2234 

236o 

2466 

2572 

2678 

106 

410 

612784 

2890 

2996 

3l02 

3207 

33i3 

3419 

3525 

363o 

3736 

106 

411 

3842 

3947 

40D3 

41 59 

4264 

4370 

4473 

458i 

4686 

4792 

106 

412 

4897 

5oo3 

5io8 

52i3 

5319 

5424 

5529 

5634 

5740 

5845 

103 

4i3 

SgDo 

6o55 

6160 

6265 

6370 

6476 

658i 

6686 

6790 

6895 

io5 

4i4 

7000 

7io5 

7210 

8257 

73i5 

7420 

7525 

7629 

7734 

7839 

7943 

io5 

4i5 

8048 

8£53 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

io5 

416 

9093 

9198 

9302 

9406 

9311 

9615 

9719 

9824 

9928 

••32 

104 

4«7 

620136 

0240 

o344 

0448 

o552 

o656 

0760 

0864 

0968 

1072 

104 

418 

1176 

1280 

1 3  84 

1488 

1592 

i6q5 

1799 

1903 

2007 

2110 

104 

419 

'   2214 

23i8 

2421 

2525 

2628 

2732 

2833 

2939 

3o42 

3i46 

104 

420 

623249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

io3 

421 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5oo4 

5io7 

52IO 

io3 

422 

53i2 

54i5 

55i8 

5621 

5724 

5827 

5929 

6o32 

6i35 

6238 

io3 

423 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

7o58 

7161 
8i85 

7263 

io3 

424 

7366 

7468 

7571 

7673 

7775 

7878 

7980 

8082 

8287 

102 

425 

8389 

8491 

8093 

8695 

8797 

8900 

9002 

9104 

9206 

9308 

102 

426 

9410 

9512 

9613 

971 5 

9817 

9919 

••21 

•123 

•224 

•326 

102 

427 

630428 

o53o 

063 1 

0733 

o835 

0936 

io38 

1139 

I24f 

1 342 

102 

428 

1444 

1 545 

1647 

1748 

1849 

1951 

2052 

2i53 

2255 

2356 

lOI 

429 

2457 

2559 

2660 

2761 

2862 

2963 

3064 

3i65 

3266 

3367 

lOI 

43o 

633468 

356q 

3670 

3771 

3872 

3973 

4074 

4175 

4276 

4376 

100 

43 1 

4477 

4578 

4679 

4779 

4880 

4981 

5o8i 

5i82 

5283 

5383 

100 

432 

5484 

5584 

5683 

5785 

5886 

5986 

6087 

6187 

6287 

6388 

loo 

433 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189 

7290 

Ih"" 

loo 

434 

7490 

7390 
8589 

7690 

7790 

7890 
8888 

7990 

8090 

8190 

8200 

8389 

99 

435 

8489 

86«9 

8789 

8988 

9088 

9188 

9287 

9387 

99 

436 

94S6 

9586 

9686 

9783 

9885 

99S4 

••84 

•i83 

•283 

•382 

99 

437 

640481 

o58i 

0680 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

99 

438 

1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 

99 

439 

2465 

2563 

2662 

2761 

2860 

2939 

3o58 

3i56 

3255 

3354 

9? 

440 

643453 

355i 

365o 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

•^o 

44 1 

4439 

4537 

4636 

4734 

4832 

4931 

5029 

5i27 

5226 

5324 

^l 

442 

5422 

5521 

56i9 

5717 

58i5 

5913 

601 1 

6110 

6208 

63o6 

9^ 

443 

6404 

65o2 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

9^ 

444 

7383 

7481 
8458 

7379 

7676 

7774 

7872 

7969 

8067 

8i65 

8262 

98 

445 

836o 

8555 

8653 

8750 

8848 

8945 

9043 

9140 

9237   97 

446 

9335 

9432 

9530 

9627 

9724 

9821 

9919 

••16 

•u3 

•210  97 

447 

65o3o8 

o4o5 

o5o2 

0399 

0696 

0793 

0890 

0987 

1084 

1181   97 

448 

1278 

1375 

1472 

1569 

1666 

1762 

1839 

1936 

2o53  2130  97 

449 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3019'  3ii6 

97 

430 

6532i3 

3309 

34o5 

35o2 

3398 

3693 

3791 

3888 

3984 

4080 

t 

431 

4177 

4273 

4369 

4465 

4562 

4658 

4754 

485o 

4946 

5o42 

^i 

432 

5i38 

5235 

533 1 

5427 

5523 

5619 

5715 

58io 

5906 

6002 

9^ 

453 

6098 

6194 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

96 

454 

7o56 

7i52 

7247 

7343 

7438 

7534 

7629 

7723 

7820  79tC)j  90 
^774!  8870  95 

455 

80!I 

8107 

8202 

8298 
9230 

1  8393 

8488 

8584 

8679 

456 

8965 

9060 

9i55 

9346 

9441 

9336 

9631 

97261  9821   95 
•676;  •771!  95 

437 

9916 

••11 

•106 

•201 

!  •296 

•391 

*4S6 

•58 1 

458 

66o865 

0960 

1033 

ii5o 

1243 

i339 

1434 

i529 

i623!  1718  95 

459 

1   j8i3 

1907 

2002 

2096 

1  2191 

2286 

1  2380 

1  2475 

;  2569J  2663   95 

0     I 

2  1  3  1  4  1  5  !  6  i  7 

1  8  i  9  i  D. 

23" 


y 


8 

A  TABLE 

OF 

LOGARITllN 

IS  FROM  1 

TO 

10,000. 

460 

0 

I 

2 

3 

4 

5 

6  1  7 

8  1  9 

D. 
94 

662708 
3701 

2852 

2947 

3o4i 

3i35 

323o 

3324i  3418 

3512'  3607 

461 

3795 

3889 

3983 

4078 

4172 

4266:  436o 

4454 

4D48 

94 

462 

4642 

4736 

4830 

4924 

5oi8 

5i  12 

5206 

5299 

5393 

5487 

94 

463 

558i 

5675 

5769 

5862 

5956 

6o5o 

6143 

6237 

633 1 

6424 

94 

464 

65i8 

6612 

6705 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

9j 

465 

7453 

7546 

7640 

7733 

7826 

7920 

8oi3 

8106 

8199 

8293 

93 

466 

8386 

8479 

8572 

8665 

8759 

8852 

8945 

903  8 

9i3i 

9224 

93 

467 

9317 

9I10 

93o3 

9596 

96S9 

9782 

9875 

9067 

••60 

•i53 

93 

468 

370246 

o339 

6431 

o524 

0617 

0710 

0802 

0895 

0988 

1080 

9? 

469 

1 173 

1265 

i358 

i45i 

1543 

1636 

1728 

1821 

1913 

2005 

93 

470 

672098 

2190 

2283 

2375 

2467 

256o 

2652 

2744 

2836 

2929 

92 

471 

302I 

3ii3 

32o5 

3297 

3390 

3482 

3574 

3666 

3758 

385o 

90 

472 

3942 

4o34 

4126 

4218 

43 10 

4402 

4494 

4586 

4677 

4769 

92 

473 

4861 

4953 

5045 

5i37 

0228 

5320 

5412 

55o3 

5595 

5687 

92 

474 

5778 

5870 

5962 

t)o53 

6145 

6236 

6328 

6419 

65ii 

6602 

92 

475 

6694 

6785 

6876 

6968 

7059 

7i5i 

7242 

7333 

7424 

7516 

91 

476 

7607 

7698 
8609 

-77  S9 
8700 

7881 

7972 
8S82 

8o63 

8i54 

8245 

8336 

8427 

91 

477 

S5i8 

8791 

8973 

9064 

9t55 

9246 

9337 

91 

47» 

9428 

9519 

9610 

9700 

979' 

9882 

9973 

••63 

•i54 

•245 

91 

479 

68o336 

0426 

o5i7 

0607 

0698 

078Q 

0879 

0970 

1060 

I  i5i 

91 

480 

681241 

i332 

1422 

i5i3 

i6o3 

1693 

1784 

1874 

1064 

2o55 

90 

481 

2145 

2235 

2326 

2416 

25o6 

2596 

2686 

2777 

2867 

2957 

90 

482 

3047 

3i37 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

483 

3947 

4o37 

4127 

4217 

4307 

4395 

4486 

4576 

4666 

4756 

90 

484 

4845 

4935 

5o35 

5ii4 

5204 

5294 

5383 

5473 

5563 

5652 

00 

485 

5742 

583 1 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

486 

6636 

6726 

68 1 5 

6904 

6994 

7083 

7172 

7261 

735i 

7440 

?9 

487 

7529 

7618 

7707 

7796 

7886 

7975 

8064 

8i53 

8242 

833 1 

\^ 

488 

8420 

85o9 

8598 

8687 

8776 

8865 

8953 

9042 

91 3 1 

9220 

«9 

489 

9309 

9398 

9486 

9575 

9664 

9753 

9841 

9930 

••19 

•107 

«9 

490 

690196 

0285 

0373 

0462 

o55o 

0639 

0728 

0816 

090D 

'^93 

89 

491 

1081 

1170 

1258 

1 347 

1435 

i524 

1612 

1700 

1789 

1877 

88 

492 

1965 

2o53 

2142 

223o 

23i8 

2406 

2494 

2583 

2671 

27D9 

88 

493 

-2847 

2935 

3o23 

3iii 

3199 

3287 

3375 

3463 

355i 

3639 

88 

494 

3727 

38i5 

3903 

3991 

4078 

4166 

4254 

4342 

443o 

4517 

88 

495 

4605 

4693 

4781 

486S 

4956 

5o44 

5i3i 

5219 

5307 

5394 

88 

496 

5482 

5569 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

t^ 

497 

6356 

6444 

653 1 

6618 

6706 

679J 

6880 

6968 

7o55 

7142 

87 

498 

7229 

7317 

74o4 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

V 

499 

8101 

8188 

S275 

8062 

8449 

8535 

8622 

8709 

8796 

8883 

87 

5oo 

698970 

9057 

9144 

9231 

93 1 7 

9404 

9491 

9578 

9664 

975i 

•617 

87 

Do  I 

9838 

9924 

©«,i 

«»«98 

•184 

*27I 

e358 

•444 

•53 1 

87 

5o2 

700704 

0790 

0877 

0963 

io5o 

ii36 

1222 

i3o9 

1395 

1482 

86 

5o3 

1 568 

1654 

1741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

86 

5o4 

243 1 

2317 

26o3 

2689 

2775 

2861 

2947 

3o33 

3ii9 

32o5 

86 

5o5 

3291 

3377 

3463 

3549 

3635 

3721 

3  Ho  7 

38^3 

3979 

4o65 

86 

5o6 

4i5i 

4s36 

4322 

4408 

449'^ 

4579 

4665 

4731 

4837 

4922 

86 

507 

5oo8 

5094 

5.79 

5265 

5350 

5436 

5522 

5607 

5693 

5778 

86 

5o8 

5864 

5949 

6o35 

6120 

6206 

6291 

6376 

6462 

6547 

6632 

85 

509 

6718 

680  3 

6888 

6974 

7059 

7144 

7229 

73 1 5 

7400 

7485 

85 

5io 

707570 

7655 

7740 

7826 

7911 

7996 

8081 

8166 

825i 

8336 

85 

5ii 

8421 

85o6 

859 1 

8676 

8761 

8846 

8931 

9015 

9100 

oi85 

85 

5l2 

9270 

9355 

9440 

9524 

9609 

9694 

9779 

9863 

9948 

••33 

85 

5i3 

710117 

0202 

0287 

0371 

0436 

0540 

0625 

0710 

0794 

0879 
1723 

85 

5i4 

0963 

1048 

Il32 

1217 

i3oi 

1 385 

1470 

1 554 

1639 

84 

1  5i5 

1807 

1892 

1976 

2060 

2144 

2229 

23 1 3 

2397 

2481 

2566 

84 

1  5i6 

265o 

2734 

2818 

2902 

2986 

3070 

3 1 54 

3238 

3323 

3407 

84 

1  5,7 

3491 

3575 

3659 

3742 

3826 

3910 

399'! 

4078 

4162 

4246 

84 

i  5i8 

433o 

4414 

4I97 

458i 

4665 

4749 

4833 

4916 

5ooo 

5o84 

84 

i  ^19 

1  N, 

5167 
0 

525i 

5335 

5418 

55o2 

5586 

5669 

5753; 

_7_l 

5836 

5920 

84 

"1j 

I 

2 

3 

_J 

5 

6 

8  ! 

9 

A  TABLE 

OF 

LOGARirii: 

,IS  FI 

iOil 

L  10 

10,000. 

g 

N. 

0 

I    2  1  3  j  4  1  5 

"~6"'j  7  1  8  1  9 

i). 

520 

716003 

6087  6170 

6254  6337  ^421 

65o4.  6588  6671!  6754 

83 

521 

6838 

6921'  7004 

7088  7171,  7254 

7338!  7421  7304 

:  7587 

11 

522 

7671 

7754  7837 

792c 

8oo3  8086 

8169I  8253  8336 
9000'  9083  9165 

8419 

83 

523 

85o2 

8585,  8668 

8751 

8834'  8917 

1 9248 

83 

534 

9331 

9414 

9497 

9580 

9663  9745 

9828'  991 1  9994 

••77 

83 

5^5 

720139 

0242 

0323 

0407 

0490  0573 

o655'  0738  0821 

0903 

83 

525 

09S6 

1 008 

u5i 

1233 

i3i6|  1398 

1481  i563  1646 

1728 

?' 

527 

1811 

189] 

1975 

2o58 

2140  2222 

23o5  2387  2469 

2552 

82 

52y 

2634 

2716 

279« 

28811  2963:  3045 

3127  3209:  3291 

3374 

82 

529 

3456 

3538 

3620 

3702!  3784 

3866 

3948  4o3o 

4II2 

4194 

82 

5Jo 

724276 

'  4358 

4440 

45221  4604 

4635 

4767  4849 

4931 

5oi3 

82 

53 1 

5095 

5176 

3258 

534o[  5422'  55o3 

5585  56671  5748 

583o 

82 

532 

5912 

5993 

6075 

6i56:  6238.  6320 

6401  6483!  6564 

6646 

82 

533 

6727 

6809 

6890 

6972}  7053 

7134 

7216  7297  7379 

7460 

8i 

534 

7  Ml 

7623 

7704 

7785;  7866 

7948 

8029  8iiO|  8191 

8273 

81 

535 

8354 

8435 

85i6 

8597!  8678 

8739 

8841 

8922  9003 

9084 

81 

536 

9165 

9246 

9327 

94081  9489 

9370 

9651 

9732  9813 

9893 

81 

537 

9974 

••55 

•i36 

•217!  ^298 

•378 

•4^9 

•540 1  •621 

•702 

81 

538 

730782 

086  3 

0944 

1024I  iio5 

1186 

1266 

1 347 

1428 

i3o8 

81 

539 

1589 

1669 

1730 

i83o 

1911 

1991 

2072 

2l52 

2233 

23i3 

81 

540 

732394 

24741  2555 

2635 

2715 

2796 

2876 

2956 

3o37 

3ii7 

80 

541 

3197 

3278  335S 

3438 

35i8 

3398 

3619 

3739 

3839 

3919 

80 

54'? 

3999 

4079  4160 

4240  4320 

4400 

4480 

4360 

4640 

4720 

80 

543 

4800 

4880  4960 

5o40|  5 1 20 

3200 

5279 

5359 

5439 

55 1 9 

80 

544 

5599 

5679  5739 

5838  3918 

5998 

6078 

6i57 

6237 

63i7 

80 

545 

6397 

6476  6556 

6635  6713 

6795 

6874 

6934 

7034 

7ii3 

80 

546 

7193 

7272  7352 

7431  73 1 1 

7390 

8463 

7749 

7829 

7908 

79 

547 

19^1 

80671  8146 

8225  83o5 

8384 

8543 

8622 

8701 

79 

548 

8781 

8860 

8939 

9018  9097 

9177 

9256 

9335 

9414 

9493 
•284 

79 

549 

9572 

9631 

9731 

9810  9889 

9968 

«®47 

•126 

•205 

79 

55o 

74o363 

0442 

0321 

0600  0678 

0757 

o836 

0913 

099  i 

1073 

79 

55i 

Il52 

I23o 

i3o9 

i388|  1467 

1546 

1624 

1703 

1782 

i860 

79 

552 

1939 

2018 

2096 

2i75|  2254 

2332 

2411 

2489 

2568 

2647 

79 

553 

2720 

2804I  2882 

2961 

3o39 

3ii8 

3196 
3980 

32751  3353 

343 1 

-8 

554 

35io 

3588  3667 

3745 

3823 

3902 

4o58 

4i36 

42i5 

78 

555 

4293 

4371  4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

78 

556 

5075 

5i5j  523' 

53o9 

5387 

5465 

5543 

5621 

5699 

5777 

78 

557 

5855 

5933  601 1 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

78 

558 

6634 

67 1 2  679O 

6868 

6945 

7023 

7101 

7179 

7236 

7334 

78 

559 

7412 

7489 

7367 

7645 

7722 

7800 

7878 

7953 

8o33 

8110 

78 

56o 

748188 

8266 

8343 

8421 

8498 

8376 

8653 

8731 

8808 

8885 

77 

56 1 

8963 

9040 

9II8 

9193 

9272 

935o 

9427 

9304 

9382 

9639 

77 

562 

9736 

98.4 

989 1 

9968 

0043 

•123 

•200 

•277 

•354 

•43 1 

77 

563 

75o5o8 

0386 

0663 

0740 

0817 

0894 

0971 

1048 

II25 

1202 

77 

564 

1279 

i356 

1433 

i5ioj  i587 

1664 

1741 

1818 

1895 

1972 

77 

565 

2048 

2125  2202 

22791  2356 

2433 

25o9 

2586 

2663 

2740 

77 

566 

2816 

2893  2970 

3o47|  3 1 23 

3200 

3277 

3353 

3430 

35o6 

77 

567 

3583 

366o!  3736 

38i3  3889 

3966 

4042 

4119 

4193 

4272 

77 

568 

4348 

4423  45oi 

4578]  4634 

4730 

4807 

4883 

4960 

5o36 

76 

569 

5lI2 

5189  3265 

5341 

5417 

5494 

5370 

5646 

5722 

5799 

76 

570 

755875 

5951 

6027 

6io3 

6180 

6236 

6332 

6408 

6484 

656o 

76 

571 

6636 

6712 

6788 

6864 

6940 

7016 

7092 

7168 

7244 

7320 

76 

572 

7396 

7472 

7548 

7624 

7700 

7775 

785i 

7927 

8oo3 

8079 

76 

573 

8i55 

823o  83o6 

8382  8458 

8533 

8609 

8685 

8761 

8836 

-6 

574 

89.2 

8988-  9o63 

9139  9214 

9290 

9366 

9441 

9517 

9392 

76 

575 

9668 

9743!  98,9 

9894  9970 

••43 

•121 

•196 

•272 

•347 

7-'> 

576 

760422 

0498,  0373 

0649  0724 

0799 

C875 

0950 

1025 

noi 

75 

577 

1 1 76 

I25i|  1326 

1402 

1477 

i552 

1637 

1702 

1778 

1 853 

75 

578 

1928 

2oo3|  2078 

2i53 

22:8 

23o3 

2378 

2453 

2529 

2604 

75 

D79 

2679 
0 

2754J  2829 

2904 

297 8  3 033 

3128 

32o3  3278! 

3353; 

75 

■    = , 

3    4  1  5  1  6 

7  1  8  1 

9  1 

D. 

^'^ 


10 

A  TABLE 

OF 

LOGARITHMS  FROM  J 

TO 

10,000. 

N. 

0 

' 

7 

3 

1  4 

5    6 

7 

8 

9 

D. 

58o 

763428 

35o3 

3578 

3653 

:  3727 

38o2  3877 

3952!  4027 

4101 

~~w 

58 1 

4176 

425i 

4326 

4400 

4475 

455o'  4624 

4699'  4774 

4848 

75 

582 

4923 

499B 

5072 j  5 147 

i  5221 

5296J  5370 

5445 

5520 

5594 

73 

583 

5669 

5743 

58 1 8 

0892 

5966 

6041 1  611 5 

6190 

6264 

6338 

74 

584 

6413 

6487 

6562 

6636 

6710 

6785 j  6859 

6933 

7007 

7082 

74 

585 

7 1 56 

723o 

7304 

7379 

7453 

75271  7601 

7675 

7749 

7823 

74 

586 

7898 

7972 

8046 

8120 

8194 

8268!  8342;  8416 

8490 

8564 

74 

^^7 

8638 

8712 

8786 

8860 

8934 

9008  9082 

9i56 

9230 

93o3 

74 

588 

9377 

945 1 

,9525 

9599 

9673 

9746  9820 

9S94 

9968 

••42 

74 

589 

770115 

0189 

0263  o336 

0410 

0484  0557 

o63i 

0705 

0778 

74 

590 

770852 

0926 

0999 

1073 

1146 

1220J  1293 

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•i85 

?7 

759 

880242 

0299  o356 

o4i3  0471  o528|  o585 

0642 

0699 

0756 

57 

N. " 

0 

1 

2 

3  !  4  1  5    6 

7  1  S 

9 

D. 

A  TABLE 

OF  LOGARITHMS  FU 

OM  1 

TO 

10,000 

13 

N. 

0 

I 

2     3 

4 

5 

6 

7 

8 

9 

D. 

760 

880814 

0S71 

0928  0985 

1042 

1099 

1 1 56 

I2l3 

1271 

i328 

57 

761 

i385 

1442 

1499'  ^556 

i6i3 

1670 

1727 

1784 

1841 

1898 

57 

762 

1955 

2012 

2069  2126 

2i83 

2240 

2297 

2354 

2411 

2468 

57 

763 

2325 

25Si 

2638;  2695 

2732 

2809 

2866 

2923 

2980 

3o37 

57 

764 

3093 

3i5o 

3207  3264 

3321 

3377 

3434 

3491 

3548 

36o5 

57 

765 

366i 

3718 

3773 

3832 

3888 

3945 

4002 

4059 

4ii5 

4172 

57 

766 

4229 

4285 

4342 

4399 

4455 

4312 

5i33 

4623 

4682 

4739 

57 

761 
76^ 

4795 

4852 

4909 

4963 

5o22 

5078 

5(92 

5248 

53o5 

57 

536i 

5418 

5474 

553 1 

5587 

5644 

5700 

5757 

58i3 

5870 

57 

769 

5926 

5983 
6547 

6039 

6096 

6i52 

6209 

6265 

6321 

6378 

6434 

56 

770 

886491 

6604  6660 

6716 

6773 

6829 

6885 

6942 

6998 

56 

771 

7054 

7111 

7167 

7223 

7280 

''l^i 

7392 

7449 

75o5 

7561 

56 

772 

7617 

7674 

7730 

7786 

7842 

7898 

85i6 

801 1 

8067 

8123 

56 

773 

8179 

8236 

8292 

8348 

8404 

8460 

8573 

8629 

8685 

56 

774 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

56 

775 

9302 

93d8 

9414 

9470 

9326 

9582 

9b38 

9694 

9750 

9806 

56 

776 

9862 

9918 

9974 

••3o 

••86 

•141 

•197 

•253 

•309 

•365 

56 

777 

890421 

°477 

0333 

o589 

0643 

0700 

0736 

0812 

0868 

0924 

56 

778 

0980 

I033 

logi 

1 147 

I203 

1259 

i3i4 

1370 

1426 

1482 

56 

779 

1637 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

56 

780 

892095 

2i5o 

2206 

2262 

23i7 

2373 

24:9 

2484 

2340 

2595 

56 

781 

265i 

2707 

2762 

2818 

2873 

2929 

298D 

3o4o 

8096 

3i5i 

56 

782 

3207 

3262 

33i8 

3373 

3429 

3484 

30  io 

3595 

365i 

3706 

56 

783 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4i5o 

42o5 

4261 

55 

784 

43i6 

4371 

4427 

44S2 

4338 

4593 

4648 

4704 

4759 

4854 

55 

■^L^ 

4870 

4925 

4980 

5o36 

5091 

5146 

5201 

5257 

53i2 

5367 

55 

786 

5423 

5478 

5533 

5588 

5644 

5699 

5734 

5809 

5864 

5920 

55 

787 

Ull 

6o3o 

6o85 

6140 

6195 

6231 

63o6 

636i 

6416 

6471 

55 

788 

658 1 

6636 

6692 

6747 

6802 

6837 

6912 

6967 

7022 

55 

789 

7077 

7i32 

7187 

7242 

7297 

7352 

7407 

7462 

1317 

7572 

55 

790 

897627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8067 

8122 

55 

791 

8176 

823i 

8286 

8341 

8396 
8944 

8451 

85o6 

856 1 

86i5 

8670 

55 

792 

8725 

8780 

8835 

8890 

8999 

9054 

9109 

9164 

9218 

55 

793 

9273 

9328 

9383 

9437 

9492 

9547 

9602 

9656 

9711 

9766 

55 

794 

9821 

9873 

9930 

9983 

•♦39 

••94 

•140 

•203 

•258 

•3l2 

55 

795 

900367 

0422 

0476 

033l 

o586 

0640 

0695 
1240 

0749 

0804 

0859 

55 

796 

0913 

0968 

1022 

1077 

ii3i 

1186 

1295 

1349 

1404 

55 

797 

1458 

I3l3 

1 567 

1622 

1676 

1731 

1783 

1840 

1894 

1948 

54 

798 

2oo3 

2037 

2112 

2166 

2221 

2275 

2320 

2384 

2438 

2492 

54 

799 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3o36 

54 

800 

903090 

3i44 

3199 

3253 

33o7 

3361 

3416 

3470 

3524 

3578 

54 

801 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

54 

802 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

54 

8o3 

4716 

4770 

4824 

4878 

4932 

4986 

5o4o 

5094 

5i48 

5202 

54 

804 

5256 

53io 

5364  5418 

5472 

5326 

558o 

5634 

5688 

5742 

54 

8o5 

6796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

54 

806 

6335 

6389 

6443 

6497 

655i 

6604 

6658 

6712 

6766 

6820 

54 

807 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

725o 

7304 

7358 

54 

808 

7411 

7463 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

54 

809 

7949 

8002 

8o56 

8110 

8i63 

8217 

8270 

8324 

8378 

843 1 

54 

810 

908485 

8539 

8592 

8646 

8699 

8753 

8807 

8860 

8914 

8967 

54 

811 

9021 

9074 

9128 

9181 

9235 

9289 

9342 

9396 

9449 

95o3 

54 

812 

9556 

9610 

9663 

9716 

9770 

9823 

9877 

9930  9984 

••37 

53 

8i3 

91 0091 

0144 

0197 

0231 

o3o4 

o358 

041 1 

0464 

03I» 

0571 

53 

814 

0624 

0678 

073 1 

0784 

o838 

0891 

0944 

0998 

io5i 

1 104 

53 

8i5 

ii58 

1211 

1264 

i3i7 

1371 

1424 

1477 

i53o 

i584 

1637 

53 

816 

1690 

1743 

1797 

i85o 

1903 

1936 

2009 

2o63 

21 16 

2169 

53 

818 

2222 

2275 

2328 

238i 

2435 

2488 

2541 

2594 

2647 

2700 

53 

2753  2806 

2839 

2913 

2066 

3019 

3072 

3i25 

3178 

323i 

53 

819 

[N. 

3284  3337 

3390 

3443 
3 

3496 
4 

3549 

36o2 

3655 

3708 

3761 

53 

0     I 

2 

5 

6 

7 

8 

9 

D- 

11 


A    TABLE    OF    LOGARITHMS     FROM     I    TO     10,000. 


N. 

0 

I  1 

2 

3 

4  1 

5  1 

^  1 

7 

8 

9 

D. 

820 

9i38i4 

3867  3920 

3973 

4026,  4070 

4i32j  4184 

4237  4290 

53 

821 

4343 

4396  4449 

45o2 

4555I  4608 

4660!  4713 

4766  4819  ^=3 
5294  5347   53 

822 

4872 

4925  4977 

5o3o 

5o83:  5 1 36 

6189!  5241 

823 

5400 

5453  55o5 

5558 

56ii|  5664 

5716J  5769 

5822  5875 

53 

824 

5927 

5980  6o33 

6o85 

6i38.  6191 

6243  6296 

6349  6401 

53 

825 

6454 

65o7  6559 

6612 

6664  6717 

6770  6822 

6875  6927 

53 

826 

6980 

7033,  7085 

7i38 

7190  7243 

72951  7348 

7400  7453 

53 

827 

7306 

755s; 

761 1 

7663 

7716:  7768 

7820  7873 

7925|  7978 

52 

828 

8o3o 

8o83 

8i35 

8188 

8240'  8293 

8345,  8397 

8450  85o2 

52 

829 

8555 

8607  8609 

8712 

87641  8816 

8869 I  8921 

8973  9026 

52 

83o 

919078 

9i3o' 

9183 

9235 

9287 1  9340 

9392!  9444 

9496  9549 

52 

83 1 

9601 

9653; 

9706 

9758 

9810,  9862 

9914  9967 

•«ig,  ••^, 

52 

832 

920123 

0176  0228 

0280 

0332; 

o384 

0436  0489 

o54i 

0593 

52 

833 

0645 

0697'  0749 

0801 

o853; 

0906 

0958  lOIO 

1062 

1114 

52 

834 

■  1166 

1218  1270 

l322 

1374; 

1426 

1478,  i53o 

i582 

i634 

-52 

835 

1686 

1738  1790 

1842 

1894I 

1946 

1998  2o5o 

2102 

2 1 54 

52 

836 

2206 

2258^  23l0 

2362 

2414 

2466 

23i8|  2570 

2622 

2674 

52 

837 

2725 

2777 

2829 

2881 

2933 

2985 

3o37|  30S9 

3i4o 

3192 

52 

838 

3244 

3296 

3348 

3399 

345i 

3do3 

3555  3607 

3658 

3710 

52 

839 

3762 

3814 

3865 

3917 

3969 

4021 

4072  4124 

4176 

4228 

52 

840 

924279 

4331 

4383 

4434 

4486 

4538 

4589  4641 

4693 

4744 

52 

841 

4796 

4848 

4899 

49^51 

5oo3 

5o54 

5io6 

5i57 

5209 

5261 

52 

842 

53 1 2 

5364 

54i5 

5467 

55i8 

5570 

5621 

5673 

5725 

5776 

52 

843 

5828 

5879  5931 

5982 

6o34 

6o85 

6i37 

6188 

6240 

6291 

5i 

844 

6342 

6394  6445 

6497 

6548 

6600 

665i 

6702 

6754 

68o5 

5i 

845 

6857 

6908 

6959 

747^ 

701 1 

7062 

7114 

7i65 

7216 

7268 

7319 

5i 

846 

7370 

7422 

7524 

7576 

7627 

7678,  7730 

7781 

7832 

5i 

847 

7883 

7935 

7986 

8037 

8088 

8140 

9191  8242 

8293 

8345 

5i 

848 

8396 

8447 

8498 

8549 

8601 

8652 

8703  8754 

88o5 

8857 

5i 

849 

8908 

8959 

9010 

9061 

9112 

9163 

92i5  9266 

9317 

9368 

5i 

85o 

929419 

9470 

9521 

9572 

9623 

9674 

9725 

9776 

9827 

9879 

5i 

85i 

9930 

9981 

••32 

••83 

•i34 

•l83 

•236 

•287 

•338 

•389 

5i 

852 

930440 

0491 

o542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

5i 

853 

0949 

1000 

io5i 

no2 

ii53 

I204 

1254 

i3o5 

i356 

1407 

5i 

854 

1438 

1 509 

i56o 

1610 

1661 

I7I2 

1763 

1814 

i865 

1915 

5i 

855 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

5i 

856 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

5i 

857 

2981 

3o3i 

3o82 

3i33 

3i83 

3234 

3285 

3335 

3386 

3437 

5i 

858 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

5i 

859 

3993 

4o44j  4094 

4145 

4195 

4246 

4296  4347 

4397 

4448 

5i 

860 

934498 

4549  4599 

465o 

4700 

.4751 

4801  4852 

4902 

4953 

5o 

861 

5oo3 

5o54  5io4 

5i54 

52o5 

5255 

53o6  5356 

5406 

5457 

5o 

862 

5507 

5558  56o8 

5658 

5709 

5759 

58o9  586o 

5910 

5960 

5o 

863 

6011 

6061 

6111 

6162 

6212 

6262 

63 13^  6363 

64 1 3 

6463 

5o 

864 

65i4 

6564 

6614 

6665 

6715 

6765 

68i5 

6865 

6916 

6966 

5o 

865 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

5o 

866 

7518 

7568 

7618 

7668 

7Ti8 

7769 

7819 

7869 

7919 

7969 

5o 

867 

8019 

8069 

8119 

8169 

8219 

8269 

832o:  8370 

8420 

8470 

5o 

868 

8520 

8570 

8620 

8670 

8720 

8770 

8820!  8870 

8920 

8970 

5o 

869 

9020 

9070 

9120 

9170 

9220 

9270 

9320  9369 

941Q 

9469 

5o 

870 

939519 

9569  9619 

9669 
0168 

9719 
0218 

9769 

9819  9869 

9918 

9968 

5o 

871 

9400 1 8 

0068 

0118 

0267 

o3i7  0367 

0417 

0467 

5o 

872 

o5i6 

o566 

0616 

0666 

0716 

0765 

081 5  o865 

0915 

0964 

5o 

873 

1014 

1064 

iii4 

ii63 

I2l3 

1263 

i3i3  i362 

1412 

1462 

5o 

874 

i5i  I 

i56i 

1611 

1660 

1710 

1760 

1809  1859 

1909 

1958 

5o 

875 

200S 

2o58 

2107 

2157 

2207 

2256 

23o6  2355 

24o5 

2455 

5o 

876 

25o4 

1  2554 

26o3 

2653 

2702 

2752 

2801  285i 

2901 

2950 

5o 

877 

3  000 

3  040 

3oQ9 

3148 

3198 

3247 

3297  3346 

3396 

3445 

59 

878 

3495 

3544  3593 

3643 

36q2 

3742 

3791  3841 

38oo 

3939 

l") 

879 

3989'  4o38  4088 

1  413-? 

4186 

4236 

4285  4335 

4384 

4433 

59 

N. 

0    I  !    2 

3  1  4    5  1  6  1  7 

1  8  1  9 

1). 

(sy 


A    TABLE    OF    LOGARITflMS    FROM     1     TO     10,000. 


15 


N.  1 

0   1  1  i 

2 

3  j  4    5    6 

7  1 

8  1  9 

49 

88o 

944483,  4532', 

458i 

463i  4680  4729'  4779 

4828;  4877;  4927 

88i 

4976  5o25 
5469,  55i8| 

5074 

5i24  5173,  5222|  5272,  532i|  5370 

5419 

49 

882 

5567 

56i6  56651 

5715;  5764'  58 1 31  5862 1 

5912 

49 

883 

596 1  j 

6010 

6059 

61081 

6157; 

6207!  6256  63o5|  6354' 

6403 

49 

884 

6452 

65oi 

655i 

6600 

6649' 

6698,  6747 i 

6796  6845i 

6894 

49 

885 

6943 

6992 

704I 

7090' 

7140; 

7189I 

7238 

7287  7336: 

7385 

49 

886 

7434 

7483 

7532 

758i 

763o 

7679  7728; 

7777'  7826: 

7875 

49 

887 

7924 

7973 

8022 

8070 

8119I 

8i68i 

8217 

8266'  83i5! 

8364 

49 

888 

8413 

8462 

85ii 

856o: 

8609! 

8657! 

8706 1 

8755I  88041 

8853 

49 

889 

8902 

8951 

8999 

90481 

9097 

9146I 

9195I 

9244  9292 

9341 

49 

890 

949390 

9439 

94S8 

9536: 

95851 

9634' 

9683; 

9731  9780. 

9829 

49 

891 

9878 

9926 

9975 

••24' 

••73 

•l2l| 

•170 

•219  ^267 

•3 16 

49 

892 

95o365 

0414 

0462 

o5iij 

o56o 

0608: 

0657 

0706 

0754' 

o8o3 

49 

893 

o85i 

0900 

0949 

0997  i 

1046 

10951 

1 143 

1192 

1240 

1289 

49 

894 

1 338 

i3S6 

1435 

1483 

i532 

i58oi 

1629 

1677 

1726 

1773 

49 

?95 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2i63 

2211 

2260 

48 

896 

23o8 

2356 

24o5 

2453 

25o2 

255o 

2599 

2647 

2696 

2744 

48 

M 

2792 

2841 

2889 

2938; 

2986 

3o34! 

3o83 

3i3i 

3i8o| 

3228 

48 

898 

3276 

3325 

3373  3421I 

3470 

35i8 

3566 

36 1 5  36631 

3711 

48 

899 

3760 

38o8 

38561  3905! 

3953 

4001 

4049 

4098  4146 

4194 

48 

900 

954243 

4291 

4339 

4387 

4435' 

4484 

4532 

458o  4628 

4677 

48 

901 

4725 

4773 

4821 

4869 

4918 

4966 

5oi4! 

5o62  5iio 

5i58 

48 

902 

5207 

5255 

53o3 

535i 

5399 

5447 

5495 

5543  5G92 

5640 

48 

903 

5688 

5736 

5784  5832 

588o 

5928 

5976 

6024 

6072 

6120 

48 

904 

6168 

6216 

6265  63i3 

636 1 

6409 

6457 

65o5 

6553 

6601 

48 

905 

6649 

6697 

6745  6793 

6840 

6888 

6936 

6984 

7082 

7080 

43 

906 

7128 

7176 

7224  7272 

7320 

7368 

7416 

7464 

7512 

7559 

48 

907 

7607 

7655 

7703 

775i 

7799 

7847 

7894 

7942 

7990 

8o38 

48 

908 

8086 

8i34 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

85i6 

48 

909 

8564 

8612 

8659  8707 

8755 

88o3 

885o 

8898 

B946 

8994 

48 

910 

959041 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

947 « 

48 

911 

9518 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

48 

912 

9995 

••42 

••90 

•i38 

•i8d 

•233 

•280 

•328 

•376 

•423 

48 

913 

960471 

o5i8 

o566 

o6i3 

0661 

0709 

0756 

0804 

o85i 

0899 

48 

914 

0946 

0994 

1041 

1089 

ii36 

1 184 

I23l 

1279 

1826 

1874 

47 

giD 

1421 

1469 

i5i6 

1 563 

1611 

1 658 

1706 

1753 

1801 

1848 

47 

916 

1895 

1943 

1990J  2o38 

2o85 

2l32 

2180 

2227 

2275 

2822 

47 

9'I 

2369 

2417 

2464;  25tl 

2559 

2606 

2653 

2701 

2748 

2795 

47 

918 

2843 

2890 

2937}  2985 

3o32 

3o79 

3i26 

3174 

3221 

3268 

47 

919 

33i6 

3363 

34ioi  3457 

35o4 

3552 

3599 

3646 

3693 

3741 

47 

920 

963788 

3835 

3382 

3929 

3977 

4024 

4071 

4118 

4i65 

4212 

47 

921 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4687 

4684 

47 

922 

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